Limits of the Form $\frac{0}{0}$

What to do when the limit is 0/0?

1. FACTORING: Factor the numerator and denomincator fully, then cancel out any common factors
2. RATIONALIZING: If it involves $\sqrt{...}\pm a$, try multiplying by the conjugate $\frac{\sqrt{...}\mp a}{\sqrt{...}\mp a}$
3. ABSOLUTE VALUES: If it involves absolute values, rewrite it without | | signs, then simplify
4. SUBSTITUTION (a.k.a. change of variable): If you can't factor the numerator and denominator, but they both involve some radical of $x$ (ex. $x^{1/a}$ and $x^{1/b}$) , let $\displaystyle u=x^{\text{LCD of 1/a \&1/b}}$

1. Factor: $\displaystyle\lim_{x\to2}\ \frac{x^2-4}{x-2}$
2. Factor: $\displaystyle\lim_{h\to0}\ \frac{4-\left(5h+2\right)^2}{h}$
3. Factor: $\displaystyle\lim_{t\to1}\ \frac{\frac{3}{t}-3}{1-t}$
4. Rationalizing: $\displaystyle\lim_{x\to1}\ \frac{\sqrt{x^2+3}-2}{x-1}$
5. Rationalizing: $\displaystyle\lim_{x\to0}\ \frac{x}{\sqrt{x+4}-\sqrt{4-x}}$
6. Absolute Values: $\displaystyle\lim_{x\to1^-}\ \frac{|x-1|}{x^2-1}$
7. Absolute Values: $\displaystyle\lim_{x\to0}\ \frac{\left|3-2x\right|-\left|2x-3\right|}{x}$
8. Absolute Values: $\displaystyle\lim_{x\to2}\ \frac{x-2}{|5x-10|}$
9. Substitution: $\displaystyle \lim_{x\to64}\ \frac{\sqrt[3]{x}-4}{\sqrt{x}-8}$
10. Substitution: $\displaystyle \lim_{x\to0}\ \frac{\left(x+27\right)^{\frac{1}{3}}-3}{x}$
11. Substitution: $\displaystyle \lim_{x\to0}\ \frac{\sqrt[6]{x+1}-1}{\sqrt{x}}$