State the squeeze theorem in words.

What are the three ingredients needed to use the squeeze theorem?

If \(-x^2\le f(x)\le x^2\) near \(0\), what is \(\lim_{x\to0}f(x)\)?

Why is \(-1\le\sin(1/x)\le1\) useful when evaluating limits near \(x=0\)?

Evaluate \(\lim_{x\to0}x^2\sin(1/x)\).

Evaluate \(\lim_{x\to0}x\sin(1/x)\).

If \(2-x^2\le f(x)\le2+x^2\), find \(\lim_{x\to0}f(x)\).

Evaluate \(\lim_{x\to0}x^4\cos(3/x)\).

Evaluate \(\lim_{x\to0}x^2\cos(1/x^2)\).

Evaluate \(\lim_{x\to0}x^2(3+\sin(1/x))\).

Explain why the inequality in the squeeze theorem only needs to hold near the approach point, not necessarily at the point itself.

Use \(|\sin u|\le1\) to show \(\lim_{x\to0}|x\sin(5/x)|=0\).

Evaluate \(\lim_{x\to0}x^2\left\lfloor\frac1{x^2}\right\rfloor\), where \(\lfloor y\rfloor\) is the greatest integer less than or equal to \(y\).

Evaluate \(\lim_{x\to0}x\left\lfloor\frac1x\right\rfloor\) as \(x\to0^+\).

Show that \(\lim_{x\to0}x^2\sin^2(1/x)=0\).

Evaluate \(\lim_{x\to0}\frac{x^2\sin(1/x)}{|x|}\).

A student tries to squeeze \(\sin(1/x)\) between \(-1\) and \(1\) and concludes \(\lim_{x\to0}\sin(1/x)=0\). Diagnose the error.

Use the squeeze theorem with \(\cos x\le\frac{\sin x}{x}\le1\) near \(0\) to find \(\lim_{x\to0}\frac{\sin x}{x}\).

Prove \(\lim_{x\to0}x\cos(1/x)=0\) using absolute values.

Suppose \(|f(x)-3|\le(x-2)^2\) for all \(x\) near \(2\). Prove \(\lim_{x\to2}f(x)=3\).