Problem

Find the limits. $$ \lim _{x \rightarrow 0}(\csc …

02:27

Problem 32 Hard Difficulty

Find the limits.
$$
\lim _{x \rightarrow+\infty}[\cos (2 / x)]^{x^{2}}
$$

Answer

$$\lim _{x \rightarrow \infty} e^{\ln \cos \left(\frac{2}{x}\right)^{x^{2}}}=e^{-2}$$

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

TOPICS IN DIFFERENTIATION

Section 6

L Hopital's Rule; Indeterminate Forms

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Video Transcript

We're looking at the limits as X goes to infinity of co sign of two of two divided by X raised to the X squared. Let's set this equal toe l And now we're gonna take the natural log of each side Natural log with limit as X goes to infinity of co sign two divided by X raised to the X squared. Now we can go ahead and switch out the limit and the natural log. Okay? And now we can go ahead and rewrite this with natural log of co sign two x on top and one over X squared on the bottom. So here we're just using our natural log property that says we could take the X squared and bring it down as a coefficient in front, and then we're rewriting it as one over X squared. So if we try to evaluate this limit, notice that in the numerator we end up getting that it tends towards zero, and in the denominator, we get it that it tends towards zero. So we have an indeterminant form of 0/0 and we are good to go to use lope atolls mobile. So let's go ahead and use low petals rule we have on the top one over co sign two divided by X, and then we also needs to multiply by the derivative of the coastline two x, which is negative. Sign two of her X times negative two or X squared just by chain rule. And then in the denominator, we gets negative two over X cubes. Okay, let's see what we get now so we can rewrite this. Simplify this expression as natural log as X goes to infinity of Let's see, Looks like it's, um, negative tan of two divided by X over one over X. And when we check this to see if we can evaluate the limit, we get 0/0, which is another indeterminant form. So we should apply Lopa tiles are a little more time, okay? And so let's go ahead and take the derivative derivative of the top. Here we get negative, too. C can't squared of tude, but by X times the derivative, the inside just negative two over x squared and then we'll divide by the derivative the denominator. So we have negative one over X squared. Okay, this simplifies 24 c can't squared of two, divided by X. Excuse me. Negative for and that gives us negative for is our limit And recall that at the start of all this we set we took a natural log. So we actually have that. The natural log of the limit that we're looking for is equal to negative for, which tells us that the limit that we're looking for a sequel to E to the negative four. Okay, and we're finally done. That's it.

We have video lessons for 84.11% of the questions in this textbook