- Find the limit of a polynomial.
- Find the limit of a power or a root.
- Find the limit of a quotient.
Evaluating Limits
The Main Idea
- Limit Laws:
- Sum/Difference: [latex]\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)[/latex]
- Product: [latex]\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)[/latex]
- Quotient: [latex]\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}[/latex], if [latex]\lim_{x \to a} g(x) \neq 0[/latex]
- Power: [latex]\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n[/latex]
- Polynomial and Rational Function Limits:
- [latex]\lim_{x \to a} p(x) = p(a)[/latex] for polynomials
- [latex]\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}[/latex] for rational functions, if [latex]q(a) \neq 0[/latex]
- Techniques for Indeterminate Forms:
- Factoring and Simplifying:
- Factor polynomials and cancel common terms
- Rationalizing (Multiplying by Conjugate):
- Used for limits with square roots
- Simplifying Complex Fractions:
- Combine fractions using LCD
- Factoring and Simplifying:
- Indeterminate Forms:
- [latex]\frac{0}{0}[/latex], [latex]\frac{\infty}{\infty}[/latex], [latex]0 \cdot \infty[/latex], etc.
- Require special techniques for evaluation
Use the limit laws to evaluate [latex]\underset{x\to 6}{\lim}(2x-1)\sqrt{x+4}[/latex]. In each step, indicate the limit law applied.
Evaluate [latex]\underset{x\to -2}{\lim}(3x^3-2x+7)[/latex].
Evaluate [latex]\underset{x\to -3}{\lim}\dfrac{x^2+4x+3}{x^2-9}[/latex]
Evaluate [latex]\underset{x\to 5}{\lim}\dfrac{\sqrt{x-1}-2}{x-5}[/latex]
Evaluate [latex]\underset{x\to -3}{\lim}\dfrac{\frac{1}{x+2}+1}{x+3}[/latex]
Evaluate [latex]\underset{x\to 3}{\lim}\left(\dfrac{1}{x-3}-\dfrac{4}{x^2-2x-3}\right)[/latex]