2.5 | Infinite Limits


Let \(f\) be a function defined on an open interval containing \(a\) with the possible of \(a\) itself. Then
$$
\lim_{x \rightarrow a} f(x) = \infty
$$
if all the values of \(f\) can be made arbitrarily large (as large as we please) by taking \(x\) sufficiently close to but not equal to \(a\). Similarly,
$$
\lim_{x \rightarrow a} f(x) = – \infty
$$
if all the values of \(f\) can be made as large in absolute value and negative as we please by taking \(x\) sufficiently close to but not equal to \(a.\)
The line \(x = a\) is a vertical asymptote of the graph of a function \(f\) if at least one of the following statements is true:
$$
\lim_{x\rightarrow a^-} f(x) = \pm \infty;
$$
$$
\lim_{x\rightarrow a^+} f(x) = \pm \infty;
$$
$$
\lim_{x\rightarrow a} f(x) = \pm \infty
$$
Let \(f\) be a function that is defined on an interval \((a, – \infty).\) Then the limit of \(f(x)\) as \(x\) approaches infinity (increases without bound) is the number \(L,\) written
$$
\lim_{x \rightarrow \infty} f(x) = L
$$
if all the values of \(f\) can be made arbitrarily close to \(L\) by taking \(x\) to be sufficiently large.
Let \(f\) be a function that is defined on the interval \((-\infty, a).\) Then the limit of \(f(x)\) as \(x\) approaches negative infinity (decreases without bound) is the number \(L,\) written
$$
\lim_{x \rightarrow – \infty} f(x) = L
$$
if all the values of \(f\) can be made arbitrarily close to \(L\) by taking \(x\) to be sufficiently large in absolute value and negative.
The line \(y = L\) is a horizontal asymptote of the graph of a function \(f\) if
$$
\lim_{x \rightarrow \infty} f(x) = L
$$
or
$$
\lim_{x \rightarrow – \infty} f(x) = L
$$
(or both).
Let \(r > 0\) be a rational number. Then
$$
\lim_{x \rightarrow \infty} \frac{1}{x^r} = 0
$$
Also, if \(x^r\) is defined for all \(x,\) then
$$
\lim_{x \rightarrow – \infty} \frac{1}{x^r} = 0
$$

E 2.5 Exercises