AP Calculus BC / Unit 1: Limits and Continuity
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
Notes
$\displaystyle \lim_{x\to\pm\infty}f(x)=L$ means $y=L$ is a horizontal asymptote.
- For rational functions: bottom degree bigger gives $y=0$; equal degrees give the ratio of leading coefficients; top degree bigger gives no horizontal asymptote.
- Growth rates: $\ln x$ grows slower than any power of $x$, which grows slower than $e^x$.
Practice
Find $\displaystyle \lim_{x\to\infty}\frac{5x^2+1}{2x^2-x}$.
Equal degrees: ratio of leading coefficients.
Find $\displaystyle \lim_{x\to\infty}\frac{3x+7}{x^3}$.
The denominator has the higher degree.
Find $\displaystyle \lim_{x\to\infty}\frac{x^{10}}{e^x}$.
Exponentials eventually outgrow any power.
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