Limit Practice

Ethan Zell

University of Michigan

Ethan Zell Limit Practice

Announcements

1 Team homework is due tomorrow at noon.

2 Note: there will be a team homework every week. The defaultdeadline is Thursday, beginning of class.

3 We will have a supervisor visit the class on Tuesday.

Ethan Zell Limit Practice

A Famous Limit

Here is a very famous limit:

limn→∞

(1 +

1

n

)n

Work with the people next to you and use estimation to make aguess at what this limit is.

In fact, this limit is e. For now, take this for granted.

Ethan Zell Limit Practice

A Famous Limit

Here is a very famous limit:

limn→∞

(1 +

1

n

)n

Work with the people next to you and use estimation to make aguess at what this limit is.

In fact, this limit is e. For now, take this for granted.

Ethan Zell Limit Practice

Another Cool Limit

Without a calculator, work with the people next to you to figureout the value of:

limn→∞

(n

n + 1

)n

Hint: Use the last slide.

limn→∞

(n

n + 1

)n

= limn→∞

(n + 1

n

)−n= lim

n→∞

(1 +

1

n

)−n.

Therefore:

= limn→∞

((1 +

1

n

)n)−1= e−1 or

1

e.

Ethan Zell Limit Practice

Another Cool Limit

Without a calculator, work with the people next to you to figureout the value of:

limn→∞

(n

n + 1

)n

Hint: Use the last slide.

limn→∞

(n

n + 1

)n

= limn→∞

(n + 1

n

)−n= lim

n→∞

(1 +

1

n

)−n.

Therefore:

= limn→∞

((1 +

1

n

)n)−1= e−1 or

1

e.

Ethan Zell Limit Practice

Another Cool Limit

Without a calculator, work with the people next to you to figureout the value of:

limn→∞

(n

n + 1

)n

Hint: Use the last slide.

limn→∞

(n

n + 1

)n

= limn→∞

(n + 1

n

)−n= lim

n→∞

(1 +

1

n

)−n.

Therefore:

= limn→∞

((1 +

1

n

)n)−1= e−1 or

1

e.

Ethan Zell Limit Practice

Group Boardwork

In groups, put the following limits into categories based on theirvalues. Draw boundaries for your categories and label them:

limx→∞

x2 limx→−∞

x2 limx→∞

ex

limx→∞

e−x limx→∞

5−x limx→∞

√x

limx→∞

ln(x) limx→∞

x−2 limx→−∞

x−2

Ethan Zell Limit Practice

Challenging Problem

Here is another famous limit:

limn→∞

n!

nn

Discuss with the people next to you what you think the answer willbe. Hint: What does the fraction look like for n = 2? What aboutn = 5? Larger?

limn→∞

n!

nn= lim

n→∞

n · (n − 1) . . . 2 · 1n · n . . . n

= limn→∞

[n − 1

n. . .

1

n

]= 0

Ethan Zell Limit Practice

Challenging Problem

Here is another famous limit:

limn→∞

n!

nn

Discuss with the people next to you what you think the answer willbe. Hint: What does the fraction look like for n = 2? What aboutn = 5? Larger?

limn→∞

n!

nn= lim

n→∞

n · (n − 1) . . . 2 · 1n · n . . . n

= limn→∞

[n − 1

n. . .

1

n

]= 0

Ethan Zell Limit Practice

Limits of Quotients

For limits of the form limx→c

f (x)g(x) , there are three types of behavior:

1 When g(c) 6= 0, you can just plug in c .

2 When g(c) = 0 but f (c) 6= 0, the limit is undefined (could be∞,−∞, or might not exist.

3 When g(c) = 0 = f (c), the limit may or may not exist andcan take any value.

Ethan Zell Limit Practice

Limits of Quotients

For limits of the form limx→c

f (x)g(x) , there are three types of behavior:

1 When g(c) 6= 0, you can just plug in c .

2 When g(c) = 0 but f (c) 6= 0, the limit is undefined (could be∞,−∞, or might not exist.

3 When g(c) = 0 = f (c), the limit may or may not exist andcan take any value.

Ethan Zell Limit Practice

Limits of Quotients

For limits of the form limx→c

f (x)g(x) , there are three types of behavior:

1 When g(c) 6= 0, you can just plug in c .

2 When g(c) = 0 but f (c) 6= 0, the limit is undefined (could be∞,−∞, or might not exist.

3 When g(c) = 0 = f (c), the limit may or may not exist andcan take any value.

Ethan Zell Limit Practice

Example of Behavior 1

limx→4

x − 2

x + 3=

4− 2

4 + 3=

2

7

Ethan Zell Limit Practice

Examples of Behavior 2

limx→10

x + 1

x − 10does not exist.

limx→10

(x + 1)2

(x − 10)2=∞.

Ethan Zell Limit Practice

Examples of Behavior 3

limx→π

(x2 − π2)

x − π= 2π.

limx→e

(x2 − e2)

(x − e)3=∞.

Ethan Zell Limit Practice

Squeeze Theorem

Theorem

If b(x) ≤ f (x) ≤ a(x) for any x close to c (except possibly x = c),and lim

x→cb(x) = L = lim

x→ca(x), then

limx→c

f (x) = L.

Ethan Zell Limit Practice

Example of Squeeze

Consider

limx→∞

7

x + e−x.

Notice that∣∣∣ 7x+e−x

∣∣∣ ≤ 7x when x > 0. We know that:

limx→∞

−7

x= 0 = lim

x→∞

7

x

So, squeeze theorem implies:

limx→∞

7

x + e−x= 0.

Ethan Zell Limit Practice

Example of Squeeze

Consider

limx→∞

7

x + e−x.

Notice that∣∣∣ 7x+e−x

∣∣∣ ≤ 7x when x > 0.

We know that:

limx→∞

−7

x= 0 = lim

x→∞

7

x

So, squeeze theorem implies:

limx→∞

7

x + e−x= 0.

Ethan Zell Limit Practice

Example of Squeeze

Consider

limx→∞

7

x + e−x.

Notice that∣∣∣ 7x+e−x

∣∣∣ ≤ 7x when x > 0. We know that:

limx→∞

−7

x= 0 = lim

x→∞

7

x

So, squeeze theorem implies:

limx→∞

7

x + e−x= 0.

Ethan Zell Limit Practice

Example of Squeeze

Consider

limx→∞

7

x + e−x.

Notice that∣∣∣ 7x+e−x

∣∣∣ ≤ 7x when x > 0. We know that:

limx→∞

−7

x= 0 = lim

x→∞

7

x

So, squeeze theorem implies:

limx→∞

7

x + e−x= 0.

Ethan Zell Limit Practice

Board Problem

(From page 81): Evaluate the following without a calculator:

limx→∞

cos2(x)

2x + 1limx→0

x4 sin(1/x)

limx→∞

x√x3 + 1

limx→∞

1

x + 2 cos2(x)

Ethan Zell Limit Practice

Exit Ticket Challenge

Mark each statement as true or false. If the statement is false,provide an example of a function which makes the statement false.

(a) If limx→0

g(x) = 0, then limx→0

f (x)g(x) =∞.

(b) If limx→0

f (x)g(x) exists, then lim

x→0f (x) exists and lim

x→0g(x) exists.

(c) If limx→c+

g(x) = 1 and limx→c−

g(x) = −1 and limx→c

f (x)g(x) exists,

then limx→c

f (x) = 0.

Ethan Zell Limit Practice