Precalculus / Unit 10: Sequences, Series and Intro to Limits
10.6 One-Sided Limits and Limits at Infinity
Notes
- $\displaystyle \lim_{x \to a^-}$ approaches from the left; $\displaystyle \lim_{x \to a^+}$ from the right. The two-sided limit exists only if both agree.
- Limits at infinity: compare degrees of the top and bottom of a rational function. Smaller top gives 0; equal degrees give the ratio of leading coefficients.
- $\displaystyle \lim_{x \to \infty}\frac{1}{x} = 0$.
- Example: $\displaystyle \frac{|x|}{x}$ is $-1$ for $x < 0$ and $1$ for $x > 0$, so $\displaystyle \lim_{x \to 0}\frac{|x|}{x}$ does not exist.
Practice
Find $\displaystyle \lim_{x \to \infty}\frac{3x^2 + 1}{x^2 - 5}$.
The degrees are equal, so the limit is the ratio of leading coefficients, $\displaystyle \frac{3}{1}$.
Find $\displaystyle \lim_{x \to \infty}\frac{5x + 2}{x^2 + 1}$.
The denominator has a higher degree.
What is $\displaystyle \lim_{x \to 0}\frac{|x|}{x}$?
The left-hand limit is $-1$ and the right-hand limit is $1$, so the limits disagree.
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