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Example Problems for the Limit of a Function

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Example I: Explain in Words What the Following Symbols Mean 0:10
  • Example II: Find the Following Limit 5:21
  • Example III: Use the Graph to Find the Following Limits 7:35
  • Example IV: Use the Graph to Find the Following Limits 11:48
  • Example V: Sketch the Graph of a Function that Satisfies the Following Properties 15:25
  • Example VI: Find the Following Limit 18:44
  • Example VII: Find the Following Limit 20:06

Transcription: Example Problems for the Limit of a Function

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

Today, we are going to do some example problems for the limit of a function.0004

Let us jump right on in.0009

The first one says, explain in words what the following symbols mean.0012

The limit as x approaches 6 of f(x) = 14, what does this mean?0016

It means the following.0023

I think I will work in red here.0026

This one says, as x gets arbitrarily close to 1,0028

I do not like arbitrarily close, let us just say closer and closer indefinitely.0037

As x gets closer and closer to 6 from below and above, 0042

because we do not have that positive or negative superscript on the number, f(x) gets closer and closer to 14.0058

That is all this symbol says.0076

As x gets arbitrarily close to 6 from below and from above, f(x) is getting close to 14 from below and from above.0078

How about this one, the limit as x approaches 4 from below of f(x) = infinity?0092

This one says, as x gets close to 4 from below, from the left, it is going to be interchangeable.0098

Sometimes, we will say from below.0114

Sometimes, we will say from the left.0115

Sometimes, we will just say left hand limit, things like that.0117

They are all synonymous.0119

As x gets close to 4 from the left, f(x) goes off to positive infinity.0121

This one, the limit as x approaches 4 from the right is equal to -3.0138

This one says, as x approaches 4 from the right, this time I will go ahead and write from above,0145

all of these things is synonymous, f(x) approaches the number -3.0169

It gets closer and closer to, approaches, all of these words are going to be used.0185

Given what you have just written, for the first limit, this one, can f(x) equal 5?0191

The limit as x approaches 6 of f(x) is 14, can f(x) = 5?0200

Absolutely, yes, it is not a problem at all because we know that the limit of a function, 0207

as it gets close to a number is independent of the value of the function at the number.0212

Yes, not a problem of all.0220

The limit as x approaches 6 of f(x).0228

It is starting to get a little crazy now with my writing.0237

Is independent of what happens when x = 6.0242

Draw all possible graph for the second limit above.0261

The second limit above that is that one right there.0268

The limit as x approaches 4 from below, this is not as x approaches -4.0274

As x approaches 4 from below, from the left of f(x) goes off to infinity.0281

We have something like this.0286

Let us say this is 4, we are going to approach 4 from below which means we are going to approach it this way.0288

We are going to take x value getting closer and closer, and then, it is going to go off to positive infinity.0296

This positive infinity, this is an asymptote.0301

This might be a graph right there.0303

As it gets closer and closer to 4, the function blows up to infinity.0308

That is it, very straightforward, just go with your intuition.0313

Clarify the following limit if it exists.0324

The limit as x approaches 0 of sin x/ x.0327

As of right now, you have two intuitive tools that you can use, in order to find the limit.0331

You have the graph of the function and you can make a table of values.0335

Taking values closer and closer to whatever number that you are approaching.0340

You can use both, and both are very powerful tools.0344

Clearly, the tabular gives you a better idea of what is happening because it gives you specific numbers.0347

But the graph is perfect because it gives you a nice first approximation for what is happening.0352

In this particular case, as x approaches 0, notice we are not specifying left hand or right hand limit.0357

We have to do both.0363

We have to approach 0 from the left from below and we have to approach it from the right from above.0365

In each case, when we look at the graph, the function looks like it is approaching 1.0373

And then from the right, the function is climbing and climbing, it looks like it approaches 1.0381

Let us go ahead and confirm that with our table of values.0386

From here to here, this is x approaching 0 from below -0.5, -0.25, 0.1, -0.01, -0.001.0389

We see it is going from 0.95, 0.9999, definitely it looks like it is getting close to 1.0402

Here x is approaching 0 from above 0.5, 0.25, 0.1, 0.01, 0.001.0413

We are getting closer and closer and closer to 0.0423

We see that the value of the function itself is going 0.95, 0.98, 0.998, 0.99999999999.0426

Yes, we can conclude that this limit is equal to 1.0437

You use your graph, you use your table of values.0446

In a subsequent lesson, we are going to learn how to do this analytically.0449

Example 3, use the graph below of f(x) to find the following limits, if they exist.0457

If they do not exist, give the reason why.0463

This is our function right here, we see it down below.0465

We have solid dot here.0469

Any place where on these graphs where you actually do not see an open dot or open, this is an open circle here.0473

It is going to be an open circle here.0480

Let me actually work in blue.0482

We see that at 3, the value of the function has actually defined what is that solid dot.0485

It is over here at 2.5.0490

The limit as x approaches 1.5 of f(x), 1.5 is here.0494

It does not specify, we have to approach it this way and approach it this way.0502

What happens to the function?0506

The function looks like it is getting close to 4.0508

From here, it looks like f is getting close to 4.0512

This limit is equal to 4, nice and simple, very straightforward.0516

The e to the left and right hand limits equal each other.0527

The limit exist and the limit = 4.0529

The limit as x approaches 3.0532

The limit as x approaches 3 from here, let us actually do c and d first because here it says 3 from below and 3 from above.0536

These two together are the same as what b is.0544

The limit as x approaches 3 from below, following this way, the y values looks like it is going to be right about there.0548

That looks like it is going to be about 3. 15, something like that.0561

Let us say this approximately equals like 3.2, and then, the limit as x approaches 3 from above.0571

When we come this way, as we get close to 3 from this way, the function is getting close to 1.0580

This one is equal to 1.0588

The left hand limit 3.20, right hand limit 1.0591

Therefore, this limit does not exist.0595

The left hand limit exists, the right hand limit exists, but because these are not equal, the limit itself does not exist at 3.0598

The limit as x approaches -2 from above.0608

Notice, this negative in front, that is the number.0612

The positive, that is from above.0615

We are approaching -2 from above.0618

What is the function doing, it looks like it is getting close to that number right there.0622

It looks like that number is about, let us say 2.1.0626

What is f(3)?0633

The value of f(3) is 2.5.0634

Notice the left hand limit is about 3.2.0641

The right hand limit, the limit from above is 1.0652

The value of the function at 3 is 2.5.0655

These are all different, they are independent.0658

You can have a left hand limit, you can have a right hand limit.0664

You are going to have the value of the function at that point.0666

They do not have to be equal.0670

They are incompletely independent.0671

If the two limits are equal, we say the limit exists.0673

If the two limits are equal and it happens that equal the value of the function, then we say that the function is continuous.0677

That is a stronger property.0685

Here, clearly, the function is discontinuous.0688

Because you see that I’m drawing my curve, I have to lift my pencil and that continue on.0690

This is a discontinuity.0694

If all three are equal which we will get to, we actually call that continuous.0695

That is a very special property, when all three are equal.0700

Let us go ahead and try another one.0707

Use the graph below of f(x) to find the following limit, if they exist.0712

Same thing that we just did, if they do not exist, give the reason why.0715

The limit as x approaches a -1.0719

-1 is here, x approaches -1, it does not specify.0726

We have to do a left hand limit and we have to do a right hand limit.0731

The left hand limit, as x approaches, we see that the function approaches this value right there, whatever that happens to be.0735

The right hand limit as we approach -1 from the right, the function is this.0751

The limit is here.0757

Clearly, this number and this number are not the same.0759

The limit as x approaches negative of f(1), it does not exist.0763

What is the value of f(-1), it is this one right here.0769

It is actually equal to 1.0775

What is the limit as x approaches 1 from below?0778

1 from below, the limit is 1.0783

The limit as x approaches 1 from above.0792

From above, the function is approaching -1.0795

A left hand limit exists, a right hand limit exists.0803

The limit itself does not exist.0806

What is the value of f(1)?0809

It turns out that the left hand limit corresponds with the value of the function at the point.0814

Notice, this is equal to that but it is not equal to this.0824

This is not continuous.0827

You can see from the graph that it is not continuous.0828

If you did not have a graph, if you just had this and this, you could say that is not continuous.0830

You do not need the graph, the limits will tell you whether something is continuous or not.0838

The limit as x approaches 3 of f(x).0844

3 is over here, we are going to approach it from below.0846

As we approach it from below, it looks like the graph is approaching -1.0852

As we approach it from above, it approach the value 3 from above, it is x does this.0858

The value of the function itself approaches -1.0864

Here, the left hand limit which is -1, right hand limit -1.0868

They equal, the limit = -1.0875

In this case, the value of the function at 3, in other words, f(3) also happens to equal -1.0877

When all three are equal, this is a continuous function.0885

As you can see, this is a nice continuous function.0888

It is perfectly smooth, in other words.0892

When we say it is continuous, we mean smooth.0894

There is no break, I do not have to lift my pencil to continue the graph.0897

This is a discontinuous function, there is a discontinuity at -1 and there is a discontinuity at +1.0902

There are left hand and right hand limits, they do not equal each other.0910

A left hand and right hand limit that do not equal each other.0914

At 3, it is continuous.0917

The left hand limit, the right hand limit, and the value of the function equal each other.0919

Sketch the graph of the function that satisfies the following properties.0926

Very simple, you just have to make a graph that satisfies these properties.0931

The limit as x approaches 0 from below is negative infinity.0937

The limit as x approaches 0 from above is positive infinity.0944

I know the graphs goes, let me do the graph in red actually.0947

Negative infinity, this one, as I approach 0 from below, my graph goes to negative infinity.0951

As I approach 0 from above, it goes to positive infinity.0957

The limit as x goes to negative infinity, as x gets really big in the negative direction is 0.0964

I know that it gets close to here, maybe not.0970

The limit as x approaches 2 of f(x) is equal to 3.0978

The limit as x approaches positive infinity is equal to 1.0990

Let us deal with this one, as x approaches positive infinity.0997

Let us stay over here on the right side of the graph, +1.1001

I’m going to go ahead and draw an asymptote at 1.1005

It tells me that as x goes to positive infinity, f(x) gets close to 1.1011

I’m going to go ahead and draw that.1016

Here as x approaches negative infinity, f(x) goes to 0.1023

I know I'm here, this is going to go probably, it is a possibility.1036

It is not the only possibility but this is the possibility.1041

f(2) is undefined.1044

As x approaches 2, it equal 3.1047

Let us say this is 2, let us say this is 3.1049

I have to actually connect it.1058

It is undefined there, there is going to be a hole there.1059

But the limit as x approaches 2, meaning from the left and from the right, the limit is the same.1063

The function actually touches here and goes down that way.1070

There we go, that is a much better function.1075

That satisfies all of the properties.1078

f(2) is undefined.1080

As x approaches 0 from below, the graph goes to negative infinity.1084

As x approaches 0 from above, the graph goes to positive infinity.1089

As x goes to negative infinity, the graph goes to 0.1093

Yes, the y value goes to 0.1096

As x approaches 2 from below, from above the graph, it approaches the number 3.1099

That takes care of that.1106

As x goes to infinity, f(x) approaches the value 1.1108

Yes, that is the horizontal asymptote and f(2) is undefined. 1112

This is a graph, not the only graph, there might be others but that is a good graph.1116

Find the following limit if it exists.1127

The limit as x approaches 0 of this function.1128

Here, we have the graph of the function.1133

We are going to use our graph to let us know what is happening as x approaches 0.1135

X approaches 0, 0 is a number.1139

It does not specify whether it is left or right hand limit, we have to do both.1142

We have to approach 0 from below, 0 from above.1146

As we approach 0 from below, the limit of f(x) as x approaches 0 from below.1152

As we approach it from below, the function, the y value goes to positive infinity.1160

The limit as x approaches 0 from above of this f(x).1169

As we go this way, the function goes to negative infinity.1173

The limit does not exist.1180

That is it, nice and simple.1183

Just use the graph to tell you what is going on.1185

If you need a table of values, you can use a table of values.1187

But here, it is clear, you can see what the graph is doing.1190

They are blowing up to infinity but in opposite directions.1194

If this 1 to infinity from the left and from the right, we can say the limit = positive infinity.1197

Determine the following limit.1211

The limit as x approaches 5 of sin x × ln x.1212

Here we avail ourselves of both the graph and the table of values.1219

From the graph, we are approaching 5.1231

This is the number we are approaching.1233

We are approaching it from this point and we are approaching if from the right.1234

From below, from above.1238

Left hand limit, right hand limit.1240

The left hand limit, as it approaches this number, we see that it gets close to some value.1244

As we approach x, we approach 5 from the right.1250

We see that the y value approaches the same value.1253

Let us see what it is at, here, x approaches 5 from below.1257

Here this is x approaching 5 from above.1264

Let us see what the numbers do.1268

4.9, 4.99, 4.999, 4.9999, getting really close to 5.1270

We notice that we are approaching -1.5433.1276

Here, same thing, we go to 5.1, 5.01, 5.001, 5.0001, getting closer and closer.1284

We see that we are approaching the same number 1.5433, -1.5433.1298

I can conclude, because the left hand limit and the right hand limit are the same, 1307

they look like they are the same number.1313

Clearly, they are the same in the graph.1315

The limit as x approaches 5 of the sin(x) × natlog(x).1317

I can say that it equal -1.543.1327

It is good to 3 decimal places.1332

That is it, very intuitive notion actually.1336

Yet, the entire modern world is based on this notion.1343

The idea of the limit, the idea of the derivative.1346

It is quite extraordinary.1348

Let us go ahead, given f(x), let us round out what we have done here today.1353

Given f(x), use your calculator or a graphing utility 1360

or graphing software to make a graph of the function that you are dealing with.1379

Make a table of values and speculate about whether the limit converges, whether the function converges to a specific number.1395

Use your graph, use your table of values and make some good choices.1434

In the next lesson, we learn how to evaluate limits analytically.1440

In other words, doing something mathematical.1467

It is going to turn out that the math is really simple.1474

Thank you for joining us here at www.educator.com.1480

We will see you next time, bye.1482