If \(u=3x\), what value does \(u\) approach as \(x\to0\)?

If \(u=x^2+1\), what value does \(u\) approach as \(x\to2\)?

Evaluate \(\lim_{x\to2}(x^2+3)^2\) using \(u=x^2+3\).

Rewrite \(\lim_{x\to0}\frac{\sin(6x)}{6x}\) using \(u=6x\).

Evaluate \(\lim_{x\to0}\frac{\sin(8x)}{x}\) using a change of variable.

Evaluate \(\lim_{x\to1}(2x-1)^5\) using \(u=2x-1\).

Evaluate \(\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}\) using \(u=\sqrt{x}\).

Evaluate \(\lim_{x\to0}\frac{1-\cos(5x)}{x}\) using \(u=5x\).

Evaluate \(\lim_{x\to1}\frac{x^3-1}{x-1}\) using \(u=x-1\).

Evaluate \(\lim_{x\to8}\frac{x^{1/3}-2}{x-8}\) using \(u=x^{1/3}\).

Evaluate \(\lim_{x\to0}\frac{\sin(2x)}{\sin(3x)}\) by changing variables in each sine factor.

Explain the new limit when \(u=g(x)\) and \(x\to a\).

Evaluate \(\lim_{x\to1}\frac{\sqrt{2x+7}-3}{x-1}\) using \(u=\sqrt{2x+7}\).

Evaluate \(\lim_{x\to0}\frac{\tan(4x)}{\sin(6x)}\).

For \(u=ax\), find \(a\) if \(\lim_{x\to0}\frac{\sin(ax)}{x}=7\).

Find \(a\) so that \(\lim_{x\to0}\frac{\sin(ax)}{\sin(5x)}=2\).

For \(u=x-a\), show that \(\lim_{x\to a}\frac{(x-a)^2}{x-a}=0\).

A student sets \(u=x^2\) in \(\lim_{x\to-2}F(x^2)\) but writes \(u\to-4\). Diagnose the error.

Explain why \(u=x^2\) can be a risky substitution for two-sided limits near \(x=0\).

Give a careful change-of-variables solution for \(\lim_{x\to0}\frac{\sin(\sqrt{x})}{\sqrt{x}}\), with the domain noted.