1.6 Limits at Infinity
Notes
Limits at infinity describe end behavior: what $f(x)$ approaches as $x$ grows very large or very negative.
- $\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0$.
- For a rational function, compare degrees. If the degree of the top is less than the bottom, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients. If the top has larger degree, the function grows without bound.
- Example: $\displaystyle \lim_{x \to \infty} \frac{3x^2 + 1}{x^2 - 5} = 3$.
- A limit of $L$ at infinity gives a horizontal asymptote $y = L$.
Practice
Find $\displaystyle \lim_{x \to \infty} \frac{4x^2 - 3x}{2x^2 + 7}$.
Equal degrees: the ratio of leading coefficients is $\displaystyle \frac{4}{2} = 2$.
Find $\displaystyle \lim_{x \to \infty} \frac{7x + 2}{x^2 + 1}$.
The bottom has the larger degree, so the fraction shrinks to 0.
What is the horizontal asymptote of $\displaystyle f(x) = \frac{6x + 1}{3x - 2}$?
The limit at infinity is $\displaystyle \frac{6}{3} = 2$.
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