Limits, The Squeeze Theorem

  • #1

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???
 

Answers and Replies

  • #2
34,140
5,762

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???
 
  • #3
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.
 
  • #4
34,140
5,762
Come on - think!
You have f(x) = x4cos(5/x), and you know that -1 <= cos(whatever) <= 1. Can't you connect the dots?
 
  • #5
yeah i worked it out =p man im a dummy thanks mark
 

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