AP Calculus AB Calculating Limits as x Goes to Infinity

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to talk about what happens when x actually goes to infinity.0004

Calculating limits, the way that we have done for previous problems.0009

Let us jump right on in.0013

We have already seen through some examples,0016

I think I’m going to work in red today.0024

I think I might work in blue.0030

We have already seen that the following is possible.0033

We have the limit as x approaches some number a of f(x) = positive or negative infinity.0045

We know that as we get close to some number, no matter what that number is,0054

that the function itself actually just blows up to infinity.0058

They go straight up or straight down to negative infinity.0061

We have seen that already.0064

For example, the limit as x approaches 3 from above, I will just leave it as 3 from below.0064

It does not matter, of 1/ x - 3 = positive infinity.0078

Let us go ahead and draw this out.0087

That is going to be something like this.0091

We know that 3 itself is not in the domain because if it were 3, it is not defined.0094

We know that we have a vertical asymptote at 3.0100

Basically what happened is, as we approach 3 from above of this function, as we approach 3,0105

this is 3 right here, from above the function just blows up to positive infinity.0112

We also know that the limit as x approaches 3 from below of this function 1/ x – 3.0119

As we approach 3 from below, coming that way, it is going to end up going to negative infinity.0127

That is all, that is all this means.0136

As x approaches some number, the function itself goes to infinity.0140

We have also mentioned the following.0146

The limit as x approaches infinity of f(x), in other words what happens when x itself goes to infinity.0159

We can take a look at this example again.0169

This 1/ x – 3, we see that as x gets really big, the function itself just gets closer and closer to 0 from above.0171

The same thing here, as we go to negative infinity, the function just gets closer and closer to the x axis.0179

In other words, to y = 0.0186

That is something that we have also seen.0188

We used graphs and tables of values to see what happens.0193

Just like in the last example, we just look at the graph to see what happens to f(x),0208

when x goes to positive or negative infinity.0220

Now we want deal with this analytically.0224

How do we actually do it mathematically?0227

We want to deal with these analytically.0235

Let us look at the example above.0246

Let us look at f(x) = 1/ x – 3.0254

We ask ourselves, what is the limit as x approaches infinity.0261

Let us go ahead and take positive infinity of the function 1/ x – 3.0273

We already saw what happens.0279

The way you handle it analytically is essentially going to be the same thing.0282

You are basically just going to put infinity in to the function and see what happens to the function.0286

That is what you are asking yourself.0291

As x gets really big, what does the whole function do?0294

You essentially do the same thing as before.0301

Just basically plugging in, instead of plugging in a number, you are plugging in infinity.0307

You essentially do the same thing as before.0315

You plug in, I mean technically, infinity it is not a number.0323

You cannot really plug it in.0327

But again, the degree of precision here is not that big a deal, we know what is happening.0329

As long as we know what is happening, we do not have to worry about pedantic little issues like this.0337

Basically, again, just plug in infinity and ask what does the function do.0342

Let me see, should I do the same thing as before?0350

Plugging in just means take x very large in the positive or negative direction and ask how the function behaves.0352

Let us do it this way, for the limit as x approaches positive infinity of 1/ x – 3.0397

When we put in infinity here, as x goes to positive infinity, the denominator gets really big.0410

The denominator gets very large.0427

Now as the denominator gets very large, what about the function itself, 1/ the denominator.0431

As the denominator gets large, the function itself gets really small.0438

Because the denominator is a bigger and bigger and bigger number.0441

1/ 1000, 1/ 100000, 1/10000000, it is going to 0.0444

As the denominator gets huge, 1/ x - 3 gets very small.0448

It approaches 0, that is what is happening.0469

That is all what we have been doing, the same thing.0474

You are essentially just going to plug it in and see what happens.0477

If you get an answer, you are good.0481

If you get something that does not make sense, you are going to have to do the same as before.0482

You are going to have to basically manipulate the expression and then take the limit again0486

as x goes to positive or negative infinity, whichever they are asking for.0489

That is the basic process over and over again.0492

We say the limit as x approaches positive infinity of 1/ x - 3 = 0.0501

The same thing with negative infinity.0511

What about the limit as x approaches negative infinity of 1/ x – 3.0519

The same thing, if we put a negative infinity in here, it is going to be a big number in the dominator.0524

It is going to be a very small function.0529

Essentially, it is going to go to 0.0532

We say it approaches 0, it just approaches it from a different point.0535

Negative infinity, these numbers in the denominator are going to be negative.0539

A positive number 1 divided by a negative number, it is going to be a negative number.0543

It is going to approach infinity from below.0546

Up here, as you go towards infinity, it approaches infinity from above.0550

That is the only difference, that vs. that.0554

They both approach 0 but from opposite ends, from the top and from the bottom.0560

Nothing strange, pretty intuitive stuff here.0573

We essentially do the same thing.0577

Sorry, I keep repeating myself.0580

We essentially do the same thing.0587

You plug in positive or negative infinity and ask what happens.0588

Again, when we say plug in positive or negative infinity, we are saying as infinity gets really big,0600

what does the function do.0605

If you get something, we essentially do the same thing, as what happens.0609

If we get something that makes sense like a number, an actual limit, that something approaches, we can stop, that is our answer.0627

We can stop, this is our answer.0643

If we get something that does not make sense, then we manipulate the expression and take the limit again.0652

Manipulate the expression and take that limit again.0683

The same limit, the limit as x goes to positive or negative infinity.0693

I wonder if I should write down what i have here next.0719

You know what, let us just launch right into the examples.0723

I think that is probably the best approach.0729

Let me go ahead and do example here.0734

We want to know what the limit is, as x approaches infinity of this rational function 7x² - 3x + 14/ 5x² + 2x + 6.0740

As x goes to infinity, the numerator goes to infinity, the denominator goes to infinity.0761

You are going to end up with this thing, infinity/ infinity.0766

This does not make sense, we are going to have to do something to this expression.0769

When you see 0/0, infinity/ infinity, infinity – infinity, because we actually do not know the rates at which these two go to infinity,0775

we do not if it actually converges to a limit or not.0783

That is why it is indeterminate.0789

You might be thinking to yourself, this one goes to infinity, this one goes to infinity, is not just 1?0790

No, it is not 1 because we do not know the rate at which these go to infinity.0794

It is indeterminate.0802

Here is how you handle a rational function.0803

For rational functions which this is, rational functions, the manipulation is the same.0806

We manipulate by dividing both the numerator and denominator by the highest power of x in the denominator.0820

In this case, the denominator, the highest power is x².0854

We are going to multiply top and bottom by x².0857

Here it is x².0863

What we get is the following.0869

We get the limit as x approaches infinity.0870

When we divide the top by x², we get 7 +,0875

Let me write up the whole thing.0884

7x² - 3x + 14/ x² divided by 5x² + 2x + 6/ x².0889

I have not changed the expression, I just multiplied the top and bottom by 1/ x².0902

This becomes the limit as x goes to infinity of 7 + 3/ x + 14x²/ 5 + 2/ x + 6/ x².0906

I just divided the x² into everything.0932

And now I take the limit again, as x goes to infinity.0935

Now I have an x in the denominator and a number on top.0938

As x goes to infinity, this goes to 0, this goes to 0, this goes to 0, this goes to 0.0941

I’m just left with 7/5, that is my limit.0948

That is how you do it.0952

Whenever you are dealing with a rational function, just divide top and bottom by the highest power in the denominator.0954

And then, take the limit again and that should give you your answer.0959

You can also have done the following with rational functions.0966

As x gets really big, whether positive or negative, basically,0971

the term that is going to dominate is going to be the highest powers in their respective polynomials.0974

In the top, the term that is going to dominate is 7x² term.0979

These basically drop out.0984

They do not contribute much.0985

In the bottom, the 5x² + 2x + 6, this term is going to dominate.0988

It drops out.0992

What you are left with is, the limit as x goes to infinity of 7x²/ 5x² which is the limit as x goes to infinity of 7/5.0994

We know that the limit of the constant is just the constant itself.1008

You can do it that way as well, absolutely fine.1011

Just ignore the lower degree of x and just worry about the highest degree x, and take it from there.1014

That is an alternate way of doing this.1023

Either one is fine, whichever works for you.1026

Let us go ahead and state the following.1030

When the limit as x goes to positive or negative infinity.1041

When you take a limit as x goes to infinity and you actually get a finite number that we got, like we got here, the 7/5, a number,1047

when a limit as x goes to infinity is an actual number, this number is a horizontal asymptote.1055

This number is a horizontal asymptote.1066

It is a horizontal, just like when we have rational functions like 1/ x – 3, the denominator,1075

the 3 is going to be a vertical asymptote because it is not defined at 3.1080

3 – 3 gives us 0.1084

When we take limits to infinity, if those limits actually give us finite numbers, those are horizontal asymptotes.1086

Let us take a look at what this particular function looks like.1094

It looks like this.1098

We see that as x goes to infinity, positive infinity or negative infinity, does not matter.1099

Let us just do it over here.1107

As x approaches positive infinity, this is the graph of the function.1114

As x gets bigger, we see that the function is actually approaching the horizontal asymptote1119

which is the dashed line, which is the 7/5.1126

This line is y = 7/5 and this of course is y = f(x), that rational function that we just worked out.1129

As x goes to positive infinity, f(x) goes to 7/5.1138

The same thing over here.1146

As x goes to negative infinity, f(x) actually approaches the same value 7/5.1148

That is all that is happening here.1156

A couple of things to notice, notice that the function drops below, crosses the asymptote,1162

and then comes up and approaches 0.1178

That is not a problem.1181

When we take limits to infinity, we are not concerned about what happens in the center of the graph.1182

We are only concerned about what happens as x gets really big, positive or negative.1187

It is not a problem for a function to actually cross a horizontal asymptote.1192

It can actually cross as many times as it likes.1198

When you deal with trigonometric functions, you might see trigonometric and exponential functions.1201

You might see that it actually crosses several times1208

but it actually gets closer and closer to some actual number to a horizontal asymptote.1211

A horizontal asymptote exists, a limit exists, but it is okay is it crosses it.1218

We are asking what happens to the function, as we get bigger and bigger.1223

In other words, is it approaching some number.1226

Do not worry about crossing the asymptote.1228

Let us write that down.1236

Crossing a horizontal asymptote is not a problem, as long as f(x) approaches some number closer and closer.1237

There you go, now we have the graphical, we have the tabular, if you need that.1270

Now we have the analytical.1276

Let us do another example here.1281

The limit as x goes to infinity 3x³ - 2x² + 4x + 14.1285

This one is easy, straight polynomial.1302

This one is easy, as x goes to positive infinity, f(x) goes to positive infinity.1310

Let us do it this way.1325

Let us deal with the highest degree.1327

As x gets really big and big, the only term that is going to dominate is going to be this term, the 3x³.1330

We do not have to worry about those.1335

As x goes to positive infinity, a positive number cubed is a positive number.1338

F(x) is going to go to positive infinity.1342

As x goes to negative infinity, this is going to dominate.1350

A negative number cubed is a negative number.1357

Therefore, f(x) itself is going to end up going towards negative infinity.1359

That is all that is going on here.1365

Notice, as x goes to positive infinity, f(x) goes to positive infinity.1376

As x goes to negative infinity, f(x) goes to negative infinity.1381

This does not actually converge to a number.1385

The functions just fly off in opposite directions.1387

This function does not converge to an actual number.1390

That can happen, we know that already.1407

We have seen limit as x approaches some number can be some number, an actual limit.1413

The limit as x approaches some number, the function can go off to infinity.1419

Now x itself can go off to positive or negative infinity.1425

The function itself can approach some number, an actual limit.1428

The limit exists, in other words.1433

Or as x goes to positive or negative infinity, the function goes to positive or negative infinity.1434

Those are the possibilities.1439

Let us do another example.1447

This is going to be slightly more involved and it is definitely an example that you want to pay close attention to.1449

The limit as x approaches infinity of 15x² + 30 all under the radical sign/ x - 1.1456

We have a rational function.1470

There is a square root on the top but it is still a rational function and some function on top/ some function on the bottom.1472

Again, a rational function, our general procedure for dealing with a rational function,1485

it implies that we divide the top and bottom by the highest degree of x in the denominator.1494

We divide the numerator and denominator by the highest power of x in the denominator.1501

Now x going to infinity means x goes to positive infinity and x goes to negative infinity.1528

I have to let you know something.1539

Some people, when you see x goes to infinity, they are saying goes to positive infinity.1541

Remember what we said in the previous lesson.1547

When you are dealing with limits as x goes to infinity, we deal with those separately.1548

x = positive infinity, x = negative infinity.1552

Here in some books and in some classes, when people say x goes to infinity, they mean positive infinity.1555

They do not mean the negative.1563

In this case, when I write x = goes to infinity, I will specify whether I mean just positive infinity, or in this case,1564

it means break it up into both its positive and negative infinity.1571

Beware of that distinction.1575

We do both.1582

Let us deal with x being greater than 0, in other words, x going to positive infinity.1587

x is greater than 0.1593

Like we said, we divide the top and bottom by the highest power.1596

We are going to get 15x² + 30 under the radical/ x/ x - 1/ x.1600

This is going to equal, x, I can think of it as √x².1611

15x² + 30, under the radical/ √x².1618

I’m just rewriting x as the √x².1625

All over here, I divide 1 - 1/ x.1629

Since this is a radical and this is a radical, I combine the radicals.1634

This is going to end up equaling 15x² + 30/ x² under the radical, /1 - 1/ x.1638

I do the actual division, now that I’m under one radical.1651

I get √15 + 30/ x²/ 1 – 1/ x.1653

This is my particular manipulation.1664

Now that I have a manipulation, now I take the limit.1667

We take the limit as x approaches positive infinity.1676

The limit as x approaches positive infinity of 15 + 30/ x² under the radical, / 1 - 1/ x.1686

As x goes to infinity, 30/ x² goes to 0 because x² blows up.1698

The denominator, it is only a constant on top, it is going to go to 0.1704

1/ x as x goes to infinity goes to 0.1708

We are left with √15.1711

√15 is our limit as x goes to positive infinity.1714

Let us deal with x less than 0.1719

X goes to negative infinity.1721

For x less than 0, it gets a little strange.1726

We still have the same thing.1737

We still have the 15x² + 30/ x, same thing, / x - 1 divided by x, dividing by the highest power of x in the denominator.1738

We are going to do the same thing that we did before.1751

Except, we are going to have 15x² + 30/ √x²/ x - 1/ x.1754

This is exactly the same thing that we did before, except for one thing, now because x is less than 0.1766

This is x and we turn it into x² as a manipulation.1775

Because x is √x², we put a negative sign here.1779

The reason we do that is the following.1783

The reason for this negative sign, I will say notice this negative sign which was not there, when we took x positive.1786

Here is why.1801

The √x², it does not actually equal x.1809

The square root of x² actually equals, we know this from algebra and pre-calculus.1814

But we do not deal with it that much, that is the problem.1821

The √x² actually equals the absolute value of x.1823

The absolute value of x is equal to x, when x is greater than 0.1828

Or it equals negative x, when x is less than 0.1835

This negative sign has to be brought in, when you are dealing with x less than 0.1839

When you turn this x into a √x², this is actually an absolute value of x.1844

That absolute value of x is a –x, that comes out here.1851

That is what is going on here.1855

I hope that makes sense.1861

It is a little strange, probably you have to think about it for a little bit but that is what is happening.1863

The reason being that the √x² is not just x.1867

It is actually the absolute value of x, because when x is negative, you can square it.1876

You can still get a positive number that you can take the square root of.1881

You have to account for x being negative or x being positive.1884

That is why this negative sign have to show up, when we are dealing with x1888

which is less than 0 because of the definition of the absolute value.1892

I hope that makes sense, in any case.1896

Now we have that expression.1899

We have -15x² + 30 under the radical, / √x²/ x - 1/ x = -15x² + 30/ x² / 1 – 1/ x.1902

All of that = -√15 - 30/ x² / 1 - 1/ x.1929

Now we take the limit.1940

The limit as x approaches -infinity of -15 -,1943

Was it – or +, I think it is + actually + 30/ x² / 1 - 1/ x.1953

As x goes to negative infinity, 30/ x² goes to 0, 1/ x goes to 0.1960

We are left with -√15 is the limit.1968

Now when we took x to positive infinity, our horizontal asymptote was √15.1975

As x goes to negative infinity, our horizontal asymptote is -√15.1984

You have two horizontal asymptotes.1989

The function is behaving differently.1991

It is approaching two different numbers, as you go big to the right and big to the left.1993

The function f(x) = the original function is 30/ x - 1 has two horizontal asymptotes.2006

y = √15 and y = -√15.2030

Let us take a look and see what this actually looks like.2038

It is going to look like this.2042

This is our function and the dashed lines are going to be the horizontal and vertical asymptotes.2043

This line right here, this is your y = √15.2051

This line right here is y = -√15.2057

I hope that makes sense.2064

Notice we also happen to have a vertical asymptote because of the x – 1 in the denominator.2067

Here, this is our vertical asymptote.2072

But as the function, here is the 0,0 mark right here.2080

As x gets really big, the function, as you can see, gets closer and closer and closer to √15.2085

As x gets really big, negative goes to negative infinity.2094

The function crosses and then comes back down, and gets closer and closer and closer and closer to -√15.2099

Whenever you see radicals in rational functions, things like this are going to happen.2108

You just have to be careful.2113

Again, let us recall that √x² actually = the absolute value of x.2114

The absolute value of x is equal to regular x, when x is greater than 0.2121

In other words, when you are going to positive infinity but it is equal to –x, when x is less than 0.2126

When you are going to negative infinity.2132

You have to have that extra negative sign.2134

It is going to confuse the heck out of you and do not worry about it.2137

To this day, I still make mistakes with stuff like this.2140

Which is why personally, I love graphs.2144

I make the graph and the graph tells me exactly what my function is doing.2147

And then, I adjust my analysis in order to fit the graph.2150

It is cheating but c'est la vie.2155

Let us do another, an example.2160

What is the limit as x approaches 0 of e¹/x.2167

This is a number, x = 0.2181

We have to do x approaches 0 from above.2183

We have to do x approaches 0 from below.2191

For x approaching 0 from above, positive numbers headed towards 0, here is what we get.2197

As x gets close to 0, this 1/x goes to positive infinity.2208

As x gets tinier and tinier, 1/ 1/10, 1/ 1/100, 1/ 1/1000000, the 1/x blows up to infinity.2220

We get e ⁺infinity = infinity.2232

The limit as x approaches 0 from above of e¹/x = positive infinity.2240

For x approaches 0 from below, as x approaches 0 from below,2253

x is a negative number that means 1/x goes to negative infinity.2269

x is a negative number.2281

It is getting closer to 0 but it is still a negative number.2283

1/x is a negative number.2288

It is going to end up going towards negative infinity.2290

1/x goes to negative infinity.2293

e ⁺negative infinity is equivalent to 1/ e ⁺infinity.2298

1/e, as this becomes infinitely large, the 1/x becomes infinitely small.2310

It goes to 0.2318

The limit as x approaches 0 from below of e¹/x is actually equal to 0.2323

You get two different limits approaching a number one from above and one from below.2333

Again, it is all based on just asking yourself what happens to the function, or in this case, pieces of the function.2339

You use that piece to address the big function, when x does something.2346

Use your intuition, trust your intuition.2351

This is completely intuitive.2354

I know the x is negative.2358

Therefore, I know that 1/x is going to be a negative number.2360

If it gets closer and closer to 0, it is going to go off to infinity but it is going to go off to negative infinity.2363

e ⁻infinity is the same as 1/ e ⁺infinity.2371

As this denominator gets big, the function itself, it goes to 0.2374

Let us take a look and see what is this.2382

In this case, the limit as x approaches 0 of e¹/x does not exist.2387

It does not exist because you ended up with two different limits.2397

One is infinite, one is 0.2400

Again, we said, when you are approaching a number,2402

the left hand limit and the right hand limit have to be the same, in order for us to say that the limit exists.2405

The limit is this.2410

Our left hand limit exists at 0, the right hand limit hand limit exists in the sense that it goes off to infinity.2415

It is not a real number but they exist, it goes off to infinity.2421

But the limit itself does not exist.2424

Let me repeat, I know you are sick to death of me repeating this, I know.2431

You see that we are simply asking ourselves,2437

what happens to f(x) as x either approaches a number or as x approaches infinity.2455

That is all we are asking.2472

The reason I keep repeating myself, I apologize, is because sometimes2473

when you are just faced with some function, it is a little intimidating symbolically.2477

You sort of all of a sudden get a little intimidated and discombobulated, just what is it asking.2481

It is saying, as x does this, what does f(x) do?2488

Just remember that is all this symbolism is.2493

As x does this, what does f(x) do?2496

As long as you contain that, you can calm down and address the issue as you need to.2499

Essentially, it is simple.2505

The hardest part in taking limits is going to be the manipulation part.2507

How do I manipulate the function, in order to actually be able to take the limit and get something that makes sense.2510

It is always going to be the hardest part of calculus.2517

It is going to be the manipulation and the algebra, not the calculus itself.2519

In any case, that is that.2523

I will write this out.2536

If our first attempt is nonsense, when we take the limit nonsense,2538

then we manipulate and try again until we get f(x) actually approaches an actual number,2549

or we get f(x) going to positive or negative infinity.2576

If it does not work the second time, when you take the limit after you manipulate it, try another manipulation.2580

Try it again, you keep trying until one of these two things happens.2584

Either you end up with an actual number, when you take the limit, or you end up with positive or negative infinity.2588

That is when you can stop.2592

Let us try, this time, the limits as x goes to positive or negative infinity of e¹/x.2598

Now it is not x approaches 0, but it is x approaches positive or negative infinity of e¹/x.2609

Let us see what happens here.2615

As x approaches positive infinity, let us go ahead and write our function again here.2623

f(x) = e¹/x, actually let me write the entire limit.2638

We wanted to know what the limit as x approaches positive or negative infinity was, of e¹/x.2645

As x approaches positive infinity, 1/x goes to 0.2651

e¹/x, e⁰ approaches 1 from above, positive numbers.2663

As x approaches negative infinity, 1/x definitely approaches 0.2675

1/x approaches 0, with 1/x being negative values.2700

e¹/x is equivalent to 1/,2710

Let me make this more clear.2727

As x goes to negative infinity, this 1/x, it goes to 0 through negative values.2728

Because if x is a negative number then 1/x is a negative number, two negative values.2735

e⁻¹/x.2749

You know what, I do not want to do that.2757

We are still dealing with e¹/x.2760

e¹/x, except now this 1/x is our negative numbers.2761

That is going to be equal to 1/ e¹/x, where now these are positive.2767

e¹/x, we said that it approaches 1, this is going to approach 1/1.2775

It is going to be 1 from below.2784

The morale of all this is just keep track of all the details.2788

Just be really meticulous in what is it that you do and everything should work out.2792

Let us take a look at what this particular thing looks like.2796

This is our function.2799

Here our function is e¹/x.2802

As x gets really big positive, we said that the function itself is going to approach 1.2806

This dashed line is our horizontal asymptote, it is y = 1.2813

As x gets really big negative, 1/x is definitely going to be negative.2820

It is the same as 1/ e¹/x, it is going to also approach 1 from below.2828

But notice it never becomes negative because this is never a negative number.2835

It approaches it from below but still positive numbers, in the sense that.2842

1/x is negative, the e¹/x which is what this function is, it approaches 1 from below.2847

In this case, there is only one horizontal asymptote.2854

It is y = 1.2857

The lesson is basically this, there is no one way to evaluate every limit.2868

That is pretty much where it comes down to.2888

This is reasonably sophisticated mathematics.2890

As things become more sophisticated in mathematics, it is no longer going to be algorithmic.2892

There is a certain degree of algorithm, or if you do this and this, each problem is going to be different.2898

You have to bring all your resources to bare, whatever those resources are.2903

Whether they be graphs, tables, analytics, some trick that you learn when you are 10 years old, whatever it happens to be.2907

All of these things have to be brought to bare, as weapons against these particular problem.2916

There is no one way to evaluate every limit.2921

Do not think that you are supposed to just look at a problem and know how to do it.2925

The process of mathematics is not looking at it and just automatically knowing what to do.2928

If you fall into that trap, then when you are faced with something that is slightly different than what you used to,2936

you are not going to able to handle it.2941

Always try to keep a reasonable degree of objectivity.2943

Keep an open mind and do not worry that all of a sudden you look at something,2945

you do not know how to do it immediately.2949

Knowing how to do something immediately is not a measure of intelligence.2952

Intelligence has to do with being able to look at a situation and work the situation out.2958

This is sophisticated mathematics, it is not 1 or 2 steps.2964

There are going to be some problems in calculus over the next year that are multiple steps.2967

You are not going to even know where it is that you are going.2972

You are going to have to trust each step you take is a reasonable step.2974

You are going to have to hope that it is taking you somewhere.2978

That is what it is about, it is about getting to the answer in a reasonable logical way.2980

Even if you do not know exactly the path that you are following.2985

It could be a very circuitous path, in any case.2988

There is no one way to evaluate every limit.2992

Use every resource at your disposal.2996

Thank you so much for joining us here at www.educator.com.2998

We will see you next time, bye.3000