This section deals with limits at infinity, that is, limits of the form $\underset{x\to \pm \infty }{lim}f\left(x\right)$, and with infinite limits, that is, limits of the form $\underset{x\to a}{lim}f\left(x\right)\=\pm \infty$.

Table 1.5.1 summarizes the terminology used in connection with "limits at infinity."

Caution:  It is tempting to say that the limit point for $\underset{x\to \infty }{lim}f\left(x\right)$ is infinity. But infinity is not a real number. The symbol ∞ that's used for this concept is just a notation for the words in Table 1.5.1. It stands for the idea that the quantity in question gets larger than any conceivable real number, but therefore is itself not a real number. That's why limits involving the notion of "infinity" require special treatment.

If the limit at infinity for $f\left(x\right)$ is a (finite) real number $L$, then the line $y\=L$ is said to be a horizontal asymptote for the graph of  $f\left(x\right)$.

If $a$ is a real number or one of the symbols $\infty$ or $-\infty$, then limits of the form $\underset{x\to a}{lim}f\left(x\right)\=\pm \infty$ are called infinite limits. If $f\to \infty$, the function values increase without bound, getting larger than any real number that can be conceived. Similarly, if $f\to -\infty$, the function values decrease without bound, getting smaller (more negative) than any real number that can be conceived.

Since the symbols $\infty$ and $-\infty$ are not themselves specific real numbers, limits such as these do not exist. This often puzzles students. On the one hand, such limits are declared to be either $\mathrm{infinity}$ or $\mathrm{negative} \mathrm{infinity}$. On the other hand, the limit does not exist as a real number. Both statements are true. The use of the symbols $\infty$ and $-\infty$ are merely shorthand notations for the concept that the function values $f\left(x\right)$ either increase or decrease without bound.

If any of the limits in Table 1.5.2 prevail, then the vertical line defined by $x\=a$ is a vertical asymptote for the graph of $f\left(x\right)$. In each case, the limit at $x\=a$ does not exist, even when the limit is said to be either of $\pm \infty$.

Each graph in Table 1.5.2 has, in addition to a vertical asymptote at $x\=a$, the horizontal asymptote $y\=0$.

Table 1.5.3 summarizes behaviors, as $x\to \+\infty$, for $x^s$, where $s$ is a real number. Figure 1.5.2 contains, for selected values of $s$, graphs of $x^s\,x\>0$.

For $s\&gt;0$, $x^s$ gets arbitrarily large as $x$ itself gets arbitrarily large. When $s\&equals;0$ and $x\&gt;0$, $x^s\&equals;x^0\&equals;1$ so that the limit, by the Identity rule, is 1. For $s<0$ and $x\&gt;0$, $x^s\&equals;x^{-\&verbar;s\&verbar;}\&equals;1\&sol;x^{-\left|s\right|}$, a fraction with a denominator that gets arbitrarily large. Hence, the limit as $x\to \infty$ in that case will be zero.

In Figure 1.5.2, the graph of $x^0\&equals;1$ is the horizontal black line. Curves above this line correspond to graphs of $x^s\&comma;s\&gt;0$; below this line, to $x^s\&comma;s<0$.

Limits at $-\infty$ for $x^s$ are more complicated. Table 1.5.4 summarizes behaviors, as $x\to -\infty$, for $x^s$, where $s$ is a real number. Figure 1.5.3 contains, for selected values of $s$, graphs of $x^s\&comma;x<0$.

The conditions in Table 1.5.4 are necessary because, for $x<0$, $x^s$ is well defined only when $s$ is an integer or a rational number. However, depending on the rational number, the behavior of $x^s$ will differ. For example, $x^{4\&sol;3}$ will tend to $\&plus;\infty$, but $x^{5\&sol;3}$ will tend to $-\infty$ because the cube root, which must be taken first, is negative, and the fourth power will make the resulting number positive, but the fifth power will keep the resulting number negative. Then again, a function like $x^{-2\&sol;3}$ or $x^{-3\&sol;5}$ will be a fraction whose denominator gets large in absolute value, and will therefore tend to zero as $x\to -\infty$. Finally, a function such as $x^{3\&sol;2}$ (the root is taken first) is not defined over the reals, so its limit to $-\infty$ cannot exist.

As in Figure 1.5.2, the horizontal black line is the graph of $x^0\&equals;1$. The dotted red and blue curves above this line are graphs of $x^{4\&sol;3}$ and $x^2$, respectively. That the graph of $x^2$ tends to $\&plus;\infty$ as $x\to -\infty$ should be no surprise. When evaluating $x^{4\&sol;3}$, the cube root is taken first, and then that number is raised to the fourth power. Hence, $x^{4\&sol;3}$ also tends to $\&plus;\infty$ as $x\to -\infty$.

The solid green and dotted orange curves just below the horizontal black line are graphs of $x^{-2\&sol;3}$ and $x^{-2}$, respectively. Since each of these functions are fractions with denominators that get large in magnitude, they both tend to zero as $x\to -\infty$.

The four graphs below the $x$-axis belong to $x^{-3}$, $x^{-1}$, $x$, and $x^3$, drawn in blue, red, green and dotted orange, respectively. The first two are fractions with denominators that get large in magnitude, so they tend to zero as $x\to -\infty$.

The graph of $x$ is the dotted green line, and the graph of $x^3$, the dash-dot orange curve, tends to $-\infty$ as $x\to -\infty$.

Figure 1.5.4 contains a graph of $x^{-2}$ while Figure 1.5.5 contains a graph of $x^{-3}$. These graphs reveal the behavior of $\underset{x\to a}{lim}{x}^s$, for $s\&equals;-2$ and $s\&equals;-3$, and $a\&equals;0$. The behavior at $a\&equals;0$ is easily translated to $x\&equals;a\ne 0$.

Each of the graphs in Figures 1.5.4 and 1.5.5 have vertical asymptotes at $x\&equals;0$. Because the functions $x^{-2}$ and $x^{-3}$ are not defined at $x\&equals;0$, the computation of any limit at $x\&equals;0$ must begin with the one-sided limits listed in Table 1.5.5.

Although each of the one-sided limits "equals" either $\pm \infty$, none exist; the symbol ∞ is not a real number. It is simply a notation for the concept that the quantity grows beyond bound. Because the one-sided limits do not exist, the two-sided limits similarly do not exist. Here again, the student is cautioned: the two-sided limit at $x\&equals;0$ for $x^{-2}$ is said to be ∞, but this limit does not exist. On the other hand, the same limit for $x^{-3}$ is declared to be "undefined" by Maple, which is Maple's way of stating "does not exist." So, neither two-sided limit exists, but for $x^{-2}$, stating that the two-sided limit is ∞ is a useful way to describe the behavior of the function on both sides of the vertical asymptote.

Table 1.5.6 gives a slightly more refined summary of the one-sided limits on either side of a vertical asymptote..

According to Table 1.5.6, the limit from the right is less complicated than the limit from the left. For ${\left(x-a\right)}^s$, the limit from the right is taken through positive real numbers, so there are only three outcomes for this limit. For example, think of $x^2\&comma;x^0\&equals;1$, and $x^{-2}$ as $x\to 0$ through the positive reals; the limiting behaviors are zero, 1, and $\&plus;\infty$.

The limit from the left is taken through the negative reals, and for these numbers ${\left(x-a\right)}^s$ is defined only for certain rational values of $s$. For example, $\underset{x\to 0-}{lim}{x}^{2\&sol;3}\&equals;\underset{x\to 0-}{lim}{x}^{5\&sol;3}\&equals;0$, but $\underset{x\to 0-}{lim}{x}^{-5\&sol;3}$=$\underset{x\to 0-}{lim}1\&sol;x^{5\&sol;3}\&equals;-\infty$, while $\underset{x\to 0-}{lim}{x}^{-2\&sol;3}\&equals;\underset{x\to 0-}{lim}1\&sol;x^{2\&sol;3}\&equals;\infty$ (because the power "2" is applied last). Of course, $\underset{x\to 0-}{lim}{x}^{3\&sol;2}$ where the square root would have to be taken first, does not exist.

If  $p_n\left(x\right)\&equals;\sum _{k\&equals;0}^n{\mathrm{\&alpha;}}_k{x}^k$ is a polynomial of degree $n$, and $q_m\left(x\right)\&equals;\sum _{k\&equals;0}^m{\mathrm{\&beta;}}_k{x}^k$ is a polynomial of degree $m$, then the rational function $f\left(x\right)\&equals;\frac{p_n\left(x\right)}{q_m\left(x\right)}$ will have asymptotes as per the conditions in Table 1.5.7.

When $n\&equals;m\&equals;c$, the rational function has for its horizontal asymptote, the line $y\&equals;\mathrm{\&lambda;}$, where $\mathrm{\&lambda;}$ is the ratio of the leading coefficients of the numerator and denominator. From Table 1.5.7, this ratio is $\mathrm{\&lambda;}\&equals;{\mathrm{\&alpha;}}_c\&sol;{\mathrm{\&beta;}}_c$.

When $n\&equals;m\&plus;1$, that is, when the degree of the numerator is one higher than that of the denominator, the rational function has for its oblique asymptote, a line of the form $y\&equals;u x\&plus;v$. A simpler prescription for the coefficients $u$ and $v$ can be written if first, the polynomials $p_n$ and $q_m$ are written as

$p\left(x\right)\&equals;a x^n\&plus;b x^{n-1}\&plus;\cdots$       

$q\left(x\right)\&equals;A x^{n-1}\&plus;B x^{n-2}\&plus;\cdots$ 

so that, $u\&equals;a\&sol;A$ and $v\&equals;\left(b A-a B\right)\&sol;A^2$, a result that is obtained by long division.$$

See also Example A-9.9 in the Appendix.

If  $p\left(x\right)\&equals;\sum _{k\&equals;0}^n{\mathrm{\&alpha;}}_k{x}^k$ is a polynomial of degree $n$, and $q\left(x\right)\&equals;\sum _{k\&equals;0}^m{\mathrm{\&beta;}}_k{x}^k$ is a polynomial of degree $m$, an algebraic technique for evaluating a limit at infinity for the rational function $p\left(x\right)\&sol;q\left(x\right)$ is to divide through both $p$ and $q$ by $x^m$. Table 1.5.8 provides schematics that lead to the evaluation of such limits at infinity.

Figure 1.5.6 is a graph of the typical periodic functions $\sin \left(x\right)$ and $\cos \left(x\right)$, both of which have periods $2 \mathrm{\&pi;}$. In every interval of length $2 \mathrm{\&pi;}$, either of these functions will take on all values between $-1$ and 1. Consequently, the limits at infinity for such functions do not exist because the function values never settle down to a unique number. Table 1.5.9 contains the Maple returns for such limits at infinity.

In Table 1.5.9, Maple returns a range for the limit at infinity of the periodic functions. This is a Maple "abbreviation" for the correct value of such limits, which is "does not exist." Some texts will shorten this phrase to some form of the acronym DNE. The proper response in a calculus class is the phrase does not exist, not the Maple range.