How to Use the Squeeze Theorem to Find This Limit?

  • MHB
  • Thread starter nycmathdad
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In summary, the Squeeze Theorem is a useful tool for finding limits in cases where direct substitution is not possible. It states that if a function is squeezed between two other functions whose limits are equal, then the limit of the squeezed function is also equal to that limit. This was demonstrated in the given conversation where the limit of x^2 • sin(1/x) was found to be 0 by squeezing it between the limits of -|x^2| and |x^2|. The same concept can be applied to other functions, as seen in the other two Squeeze Theorem proof threads.
  • #1
nycmathdad
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Use the Squeeze Theorem to find the limit.

lim (x^2 • sin(1/x))
x--> 0

Let me see.

-1 ≤ sin (1/x) ≤ 1

-x^2 ≤ x^2 • sin(1/x) ≤ x^2

-|x^2| ≤ x^2 • sin(1/x) ≤ |x^2|

lim -|x^2| as x tends to 0 = 0.

lim |x^2| as x tends to 0 = 0.
.
By the Squeeze Theorem, x^2 • sin(1/x) was squeezed between the limit of -|x^2| as x tends to 0 and the limit of |x^2| as x tends to 0.

Conclusion:

lim (x^2 • sin(1/x)) = 0
x--> 0

The limit is 0.

Correct?
 
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  • #2
Yes, that is correct.
 
  • #3
Country Boy said:
Yes, that is correct.

Very cool. What about the other two Squeeze Theorem Proof threads?
 

FAQ: How to Use the Squeeze Theorem to Find This Limit?

What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem, is a mathematical theorem used to prove limits. It states that if two functions, g(x) and h(x), are both approaching the same limit as x approaches a certain value, and a third function, f(x), is always between g(x) and h(x), then f(x) must also approach the same limit as x approaches that value.

How is the Squeeze Theorem used in proofs?

The Squeeze Theorem is used to prove limits by showing that a function is "sandwiched" between two other functions with known limits. By using the Squeeze Theorem, we can determine the limit of the middle function without knowing its exact value.

What are the requirements for using the Squeeze Theorem in a proof?

In order to use the Squeeze Theorem in a proof, the three functions (g(x), h(x), and f(x)) must all be defined and continuous on an interval containing the value that x is approaching. Additionally, the limit of g(x) and h(x) must be the same as x approaches the value.

Can the Squeeze Theorem be used to prove all limits?

No, the Squeeze Theorem can only be used to prove limits when the function is "sandwiched" between two other functions. If the function is not bounded by two other functions, then the Squeeze Theorem cannot be used.

Are there any limitations to the Squeeze Theorem?

Yes, the Squeeze Theorem can only be used to prove limits as x approaches a specific value. It cannot be used to prove limits at infinity or for limits involving oscillating functions.

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