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.
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).
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.
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.
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.
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.