Limit Laws and Algebraic Techniques
Apply the sum, product, quotient, and composition limit laws. Evaluate limits by direct substitution, factoring, rationalizing, and applying the squeeze theorem to prove trigonometric limits.
A lesson of AP Calculus AB, in Limits and Continuity, about 10 minutes.
Limit Laws and Algebraic Evaluation
By the end of this lesson, you will be able to apply the limit laws to evaluate limits algebraically and identify the appropriate technique for different forms.
The Limit Laws
If $\lim_{x\to a}f(x) = L$ and $\lim_{x\to a}g(x) = M$, then:
| Law | Formula |
|---|---|
| Sum | $\lim[f+g] = L+M$ |
| Difference | $\lim[f-g] = L-M$ |
| Product | $\lim[fg] = LM$ |
| Quotient | $\lim[f/g] = L/M$ (if $M \neq 0$) |
| Constant multiple | $\lim[cf] = cL$ |
| Power | $\lim[f^n] = L^n$ |
| Root | $\lim[\sqrt[n]{f}] = \sqrt[n]{L}$ (if defined) |
Direct substitution property: for polynomials and rational functions (when denominator ≠ 0), just plug in $a$.
Techniques for Indeterminate Forms
When direct substitution gives $\frac{0}{0}$, try:
Factor and cancel: $\lim_{x\to 3}\frac{x^2-9}{x-3} = \lim_{x\to 3}\frac{(x-3)(x+3)}{x-3} = \lim_{x\to 3}(x+3) = 6$
Multiply by the conjugate (for radicals): $\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x} = \lim_{x\to 0}\frac{(\sqrt{x+4}-2)(\sqrt{x+4}+2)}{x(\sqrt{x+4}+2)} = \lim_{x\to 0}\frac{x}{x(\sqrt{x+4}+2)} = \frac{1}{4}$
Simplify complex fractions: $\lim_{x\to 0}\frac{\frac{1}{x+2}-\frac{1}{2}}{x} = \lim_{x\to 0}\frac{\frac{2-(x+2)}{2(x+2)}}{x} = \lim_{x\to 0}\frac{-1}{2(x+2)} = -\frac{1}{4}$
Summary
Limit laws: apply to sums, products, quotients; let you break complicated limits into simpler ones.
Direct substitution: works for polynomials and rationals (non-zero denominator).
$\frac{0}{0}$ forms: factor/cancel, rationalize (multiply by conjugate), or simplify algebraically.
Always simplify first, then substitute.