Limits at Infinity

Limits at Infinity and Horizontal Asymptotes

Limits at Infinity

So far we have studied limits as $x \to a^+$ , $x \to a^-$ and $x \to a$ . Now we will consider what happens as '' $x \to \infty$ '' or '' $x \to -\infty$ ". What does that mean?

$\displaystyle{\lim_{x \to \infty}f(x)}$ describes what happens to $f$ when $x$ grows without bound in the positive direction. The word ''infinity'' comes from the Latin "infinitas", which stands for "without end" (in=not, finis=end). Imagine taking bigger and bigger values of $x$ , like a hundred, a thousand, a million, a billion, and so on, and seeing what $f(x)$ does. For instance, the statement $\displaystyle{\lim_{x \to \infty}} f(x)=7$ means that, as $x$ grows larger and larger, $f(x)$ is closer and closer to 7. We call the line $y=7$ a horizontal asymptote of $f$ , since as $x$ grows larger and larger, $f(x)$ starts looking like the line $y=7$ .

$\displaystyle{\lim_{x \to -\infty}f(x)}$ is similar, but in the negative direction. Look at $x$ being minus a million, minus a billion, minus a trillion, etc.

If $\displaystyle{\lim_{x \to -\infty} f(x) = 3}$ , then the graph of $y=f(x)$ will be very close to the horizontal line $y=3$ when $x$ is large and negative. Then the line $y=3$ is a horizontal asymptote of $f$ .

Horizontal Asymptotes

Definition: The line

$y=L$ is called a horizontal asymptote for

$y=f(x)$ if and only if

$\lim_{x\to\infty}f(x)=L, \quad \text{ or }\quad \lim_{x\to-\infty}f(x)=L$

Can a function have more than two horizontal asymptotes?

For instance, the graph on the left has both $y=\pi/2$ and $y=-\pi/2$ as horizontal asymptotes. The one on the right has horizontal asymptotes $y=\pm 4$ .