Is the Squeeze Theorem Correctly Applied Here?

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In summary, the given conversation discusses how to prove that the limit of x^2 • f(x) is equal to 0 given that 0 ≤ f(x) ≤ 1 for every x. The proof involves multiplying both sides of the inequality by x^2 and using the squeeze theorem to show that the limit is indeed 0. It is also mentioned that the fact that x^2 is always greater than or equal to 0 is necessary for the proof to hold true.
  • #1
nycmathdad
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If 0 ≤ f(x) ≤ 1 for every x, show that
lim [x^2 • f(x)] = 0.
x--> 0

Let me see.

0 ≤ f(x) ≤ 1

Multiply all terms by x^2.

0 • x^2 ≤ x^2• f(x) ≤ 1 • x^2

0 ≤ x^2 • f(x) ≤ x^2

Is this right so far? If correct, what's next?
 
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  • #2
Hi nycmathdad

Looks good. To conclude the proof, apply the squeeze theorem to the inequality $0\leq x^{2}f(x)\leq x^{2}$ you derived.

One small note: for your argument to be 100% buttoned-up, you should mention that $x^{2}\geq 0$ for all $x\in\mathbb{R}$. This is necessary to ensure that the implication $0\leq f(x)\leq 1\,\Longrightarrow\, 0\leq x^{2}f(x)\leq x^{2}$ really does follow.
 
  • #3
GJA said:
Hi nycmathdad

Looks good. To conclude the proof, apply the squeeze theorem to the inequality $0\leq x^{2}f(x)\leq x^{2}$ you derived.

One small note: for your argument to be 100% buttoned-up, you should mention that $x^{2}\geq 0$ for all $x\in\mathbb{R}$. This is necessary to ensure that the implication $0\leq f(x)\leq 1\,\Longrightarrow\, 0\leq x^{2}f(x)\leq x^{2}$ really does follow.

Ok. Thanks.
 

FAQ: Is the Squeeze Theorem Correctly Applied Here?

1. What is the Squeeze Theorem?

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a mathematical theorem that states if two functions, f(x) and g(x), are both "squeezing" another function h(x) between them, and the limits of f(x) and g(x) are equal as x approaches a certain value, then the limit of h(x) at that value also exists and is equal to the limits of f(x) and g(x).

2. How is the Squeeze Theorem used in proofs?

The Squeeze Theorem is used in proofs to show that a given function has a specific limit at a certain value. By finding two functions that "squeeze" the given function and have equal limits, the Squeeze Theorem allows us to conclude that the given function also has that same limit at that value.

3. What are the conditions for the Squeeze Theorem to apply?

In order for the Squeeze Theorem to apply, the two "squeezing" functions, f(x) and g(x), must have the same limit as x approaches the value in question. Additionally, the function h(x) being squeezed between f(x) and g(x) must also exist and have a limit at that value.

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

Yes, the Squeeze Theorem can be used to evaluate limits at infinity. In this case, the two "squeezing" functions, f(x) and g(x), must approach the same value as x approaches infinity, and the function h(x) must also exist and approach that same value as x approaches infinity.

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

The Squeeze Theorem has many real-life applications, particularly in physics and engineering. It can be used to prove the existence of limits in physical systems, such as the acceleration of an object under the influence of multiple forces. It is also used in real analysis to prove the convergence of sequences and series.

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