If \(\lim_{x\to a}f(x)=4\) and \(\lim_{x\to a}g(x)=-1\), find \(\lim_{x\to a}(f(x)+g(x))\).

State the condition needed before applying \(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}\).

If \(\lim_{x\to2}f(x)=5\), find \(\lim_{x\to2}3f(x)\).

If \(\lim_{x\to1}h(x)=2\), find \(\lim_{x\to1}(h(x))^4\).

Evaluate \(\lim_{x\to3}(2x^2-5x+1)\).

Evaluate \(\lim_{x\to4}\sqrt{x+5}\).

Evaluate \(\lim_{x\to2}\frac{x^2+1}{x+3}\).

Evaluate \(\lim_{x\to-1}(x^3+2)(4-x)\).

Given \(\lim f=2\), \(\lim g=-3\), evaluate \(\lim(4f^2-g)\) at the same approach point.

Evaluate \(\lim_{x\to1}\frac{x^2-1}{x-1}\) by using limit laws after simplification.

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

Explain why the quotient law cannot be applied directly to \(\lim_{x\to2}\frac{x^2-4}{x-2}\).

Evaluate \(\lim_{x\to9}\frac{\sqrt{x}-3}{x-9}\).

Evaluate \(\lim_{x\to1}\frac{x^3-1}{x^2-1}\).

Find \(c\) so that \(\lim_{x\to2}\frac{x^2+cx-6}{x-2}\) is finite.

For what values of \(a\) is \(\lim_{x\to1}\sqrt{a+x}\) a real limit using the root law?

Determine \(k\) if \(\lim_{x\to3}(kx^2-2x)=30\).

A student writes \(\lim_{x\to0}\frac{x+1}{x}=\frac{1}{0}=0\). Diagnose the mistake.

Show that if \(\lim f=L\) and \(\lim g=0\), the product limit \(\lim(fg)=0\), provided \(L\) exists and is finite.

Explain why limit laws cannot prove \(\lim_{x\to0}\sin(1/x)\) exists.