AP Calculus BC / Unit 1: Limits and Continuity
1.5 Determining Limits Using Algebraic Properties of Limits
Notes
Limits follow algebra rules when each piece has a limit: the limit of a sum, difference, product, constant multiple or quotient (if the bottom isn't 0) is the sum, difference, product and so on of the limits.
- $\displaystyle \lim [f(x)+g(x)] = \lim f(x) + \lim g(x)$
- $\displaystyle \lim [f(x)g(x)] = \lim f(x) \cdot \lim g(x)$
- $\displaystyle \lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$ if $\displaystyle \lim g(x)\ne0$
- For polynomials and other continuous functions, just substitute.
Practice
If $\displaystyle \lim_{x\to3} f(x)=4$ and $\displaystyle \lim_{x\to3} g(x)=-2$, what is $\displaystyle \lim_{x\to3}[2f(x)-g(x)]$?
$2(4)-(-2)=10$.
Using the same limits, what is $\displaystyle \lim_{x\to3} \frac{f(x)}{g(x)}$?
$\displaystyle \frac{4}{-2}=-2$, and the bottom limit isn't 0.
$\displaystyle \lim_{x\to -1}(x^3+4x)$ equals:
Substitute: $-1-4=-5$.
Ready for more? Take the Unit 1 test.