Is the Squeeze Theorem Correctly Applied Here?

[nycmathdad]

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?

 

[GJA]

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.

 

[nycmathdad]

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.