AP Calculus AB / Unit 1: Limits and Continuity
1.6 Determining Limits Using Algebraic Manipulation
Notes
If substitution gives $\displaystyle \frac{0}{0}$, rewrite the expression first.
- Factor and cancel: $\displaystyle \frac{x^2-1}{x-1}=x+1$ for $x\ne1$.
- Rationalize with the conjugate when there are square roots.
- Simplify complex fractions by multiplying by a common denominator.
Practice
Find $\displaystyle \lim_{x\to5}\frac{x^2-25}{x-5}$.
$(x-5)(x+5)/(x-5)=x+5\to10$.
Find $\displaystyle \lim_{x\to0}\frac{\sqrt{x+9}-3}{x}$.
Multiply by the conjugate: $\displaystyle \frac{1}{\sqrt{x+9}+3}\to\frac{1}{6}$.
Find $\displaystyle \lim_{x\to2}\frac{\frac{1}{x}-\frac{1}{2}}{x-2}$.
$\displaystyle \frac{2-x}{2x(x-2)}=-\frac{1}{2x}\to-\frac{1}{4}$.
Ready for more? Take the Unit 1 test.