What should you try first when evaluating an algebraic limit?

What indeterminate form often signals that factoring or rationalizing may be needed?

Evaluate \(\lim_{x\to2}(3x^2-5x+1)\).

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

Evaluate \(\lim_{x\to2}\frac{x^2-4}{x-2}\).

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

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

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

Evaluate \(\lim_{x\to3}\frac{x^2-9}{x^2-5x+6}\).

Evaluate \(\lim_{h\to0}\frac{(2+h)^2-4}{h}\).

Evaluate \(\lim_{x\to0}\frac{(x+1)^3-1}{x}\).

Evaluate \(\lim_{x\to2}\frac{x^3-8}{x-2}\).

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

Evaluate \(\lim_{x\to8}\frac{x^{2/3}-4}{x-8}\) using \(u=x^{1/3}\).

Evaluate \(\lim_{x\to1}\frac{x^3-1}{x^2-1}\), carefully stating what cancels.

Evaluate \(\lim_{x\to0}\frac{\frac1{x+1}-1}{x}\).

A student cancels \(x\) in \(\frac{x+2}{x}\) and gets \(2\). Explain why this is invalid, then describe \(\lim_{x\to0}\frac{x+2}{x}\).

Diagnose the error: direct substitution in \(\lim_{x\to2}\frac{x^2-4}{x-2}\) gives \(\frac00\), so the limit is \(0\).

Explain why canceling a factor in a limit problem does not require the original function to be defined at the approach point.

Evaluate \(\lim_{h\to0}\frac{\sqrt{a+h}-\sqrt a}{h}\) for \(a>0\).