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

In summary, the given limit is undefined when x=0, but the squeeze theorem can be used to find two functions, g(x) and h(x), that bound the function f(x)=x^4cos(5/x). By recognizing that -1 ≤ cos(θ) ≤ 1, we can use this to find g(x) and h(x) and apply the squeeze theorem to solve the problem.
  • #1
steve snash
50
0

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?
 
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  • #2
steve snash said:

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.
steve snash said:
(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)?
steve snash said:
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
Mark44 said:
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
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 I am a dummy thanks mark
 

1. What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a mathematical tool used to evaluate limits. It states that if two functions, g(x) and h(x), "squeeze" a third function, f(x), between them for values of x close to a certain point, then the limit of f(x) as x approaches that point is equal to the limit of g(x) and h(x) at that point.

2. How is the Squeeze Theorem used to evaluate limits?

The Squeeze Theorem is used to evaluate limits by finding two functions that "squeeze" the given function between them, and then evaluating the limits of those two functions at the same point. If the limits of the two functions are equal, then the limit of the given function at that point can also be determined to be equal. This is especially useful when the given function is complex or difficult to evaluate directly.

3. What are the conditions for the Squeeze Theorem to be applicable?

In order for the Squeeze Theorem to be applicable, the two "squeezing" functions, g(x) and h(x), must have the same limit at the point being evaluated. Additionally, the given function, f(x), must be sandwiched between g(x) and h(x) for values of x close to the point being evaluated.

4. Can the Squeeze Theorem be used to evaluate limits at infinity?

Yes, the Squeeze Theorem can also be used to evaluate limits at infinity. In this case, the two squeezing functions, g(x) and h(x), must have the same limit at infinity and the given function, f(x), must be sandwiched between g(x) and h(x) for large values of x. The limit of g(x) and h(x) can then be used to determine the limit of f(x) at infinity.

5. What are some real-life applications of the Squeeze Theorem?

The Squeeze Theorem has various applications in fields such as physics, engineering, and economics. For example, it can be used to analyze the behavior of a system approaching equilibrium, to evaluate the accuracy of measurement devices, or to determine the growth rate of a population. In finance, the Squeeze Theorem can be used to analyze the risk involved in different investment strategies.

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