How can the squeeze theorem be used to find g(x) and h(x) for a given limit?

[steve snash]

Homework Statement

Given the limit
lim(x→0) x^4 cos(5/x)
use the squeeze theorem to find g(x) and h(x) given,
f(x)=x^4 cos(5/x)

g(x)≤f(x)≤h(x)

The Attempt at a Solution

well the limit with x=0 substituted in would mean the limit is undefined as,
0^4 cos(5/0)
(5/0) can not occur but due to the function being trigonometric the usual use of 0^- and 0^+ can not be used, i found as -1≤cos θ≤1

i have no idea how to find the g(x) and h(x) with only an f(x) function, can anyone please talk me through how to use the squeeze theorem for this problem?

 

[Mark44]

Homework Statement

Given the limit
lim(x→0) x^4 cos(5/x)
use the squeeze theorem to find g(x) and h(x) given,
f(x)=x^4 cos(5/x)

g(x)≤f(x)≤h(x)

The Attempt at a Solution

well the limit with x=0 substituted in would mean the limit is undefined as,
0^4 cos(5/0)

The whole idea of limits is to be able to determine the behavior of a function that is undefined at some point. Except in the simplest cases, where a limit is unnecessary anyway, you NEVER just substitute the limiting x value into the function.

Just because x⁴ cos(5/x) is undefined at 0 doesn't mean that the limit doesn't exist. For example (sin x)/x is undefined at x = 0, yet the limit of this function as x approaches 0 does exist, and is in fact equal to 1.

(5/0) can not occur but due to the function being trigonometric the usual use of 0^- and 0^+ can not be used, i found as -1≤cos θ≤1

You have the seed of an idea here, since -1 <= cos(whatever) <= 1. How can you apply this idea to your problem to find two functions that bound x⁴ cos(5/x)?

i have no idea how to find the g(x) and h(x) with only an f(x) function, can anyone please talk me through how to use the squeeze theorem for this problem?

 

[steve snash]

You have the seed of an idea here, since -1 <= cos(whatever) <= 1. How can you apply this idea to your problem to find two functions that bound x⁴ cos(5/x)?

yes i know that i can use this to find the lower and upper (gx and hx) but my problem is how do i use this (-1<=cos theta<=1) to find the gx and hx??

can someone walk me through how to use this knowledge, where do i start, i need to know this for study purposes.

 

[Mark44]

Come on - think!
You have f(x) = x⁴cos(5/x), and you know that -1 <= cos(whatever) <= 1. Can't you connect the dots?

 

[steve snash]

yeah i worked it out =p man I am a dummy thanks mark