AP Calculus AB / Unit 1: Limits and Continuity
1.14 Connecting Infinite Limits and Vertical Asymptotes
Notes
If $f(x)\to\pm\infty$ as $x\to c^-$ or $x\to c^+$, the line $x=c$ is a vertical asymptote.
- For rational functions, look for zeros of the denominator that don't cancel.
- Check signs on each side to decide $+\infty$ or $-\infty$.
Practice
Find $\displaystyle \lim_{x\to2^-}\frac{1}{x-2}$.
For $x$ slightly less than 2, the denominator is a tiny negative number.
Where does $\displaystyle f(x)=\frac{x-1}{x^2-1}$ have a vertical asymptote?
$\displaystyle \frac{x-1}{(x-1)(x+1)}$: $x=1$ cancels (a hole), $x=-1$ doesn't.
$\displaystyle \lim_{x\to0}\frac{1}{x^2}$ is:
$x^2$ is positive on both sides, so both sides go to $+\infty$.
Ready for more? Take the Unit 1 test.