Evaluate \(\lim_{x\to0}\frac{\sin x}{x}\) using L'Hopital's rule, after checking the form.

Evaluate \(\lim_{x\to0}\frac{e^x-1}{x}\) using L'Hopital's rule.

Evaluate \(\lim_{x\to1}\frac{x^2-1}{x-1}\) using L'Hopital's rule.

Evaluate \(\lim_{x\to\infty}\frac{x}{e^x}\) using L'Hopital's rule.

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

Evaluate \(\lim_{x\to0}\frac{\ln(1+x)}{x}\).

Evaluate \(\lim_{x\to\infty}\frac{\ln x}{x}\).

Evaluate \(\lim_{x\to0}\frac{x-\sin x}{x^3}\).

Explain why L'Hopital's rule should not be applied directly to \(\lim_{x\to1}\frac{x+2}{x+3}\), and evaluate the limit.

Evaluate \(\lim_{x\to0^+}x\ln x\) by rewriting it in a form suitable for L'Hopital's rule.