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Section 2.6 Limits Involving Infinity; Asymptotes of Graphs

Note: \(\infty\) is not a number!

Definition 2.6.1. Limits at infinity.

We write \(\displaystyle\lim_{x\rightarrow\infty} f(x)=L\text{,}\) the limit as \(x\) approaches infinity of \(f(x)\) is \(L\text{,}\) if the value of \(f(x)\) is arbitrarily close to \(L\) for \(x\) sufficiently large.

Likewise, \(\displaystyle\lim_{x\rightarrow -\infty} f(x)=L\text{,}\) the limit as \(x\) approaches negative infinity of \(f(x)\) is \(L\text{,}\) if the value of \(f(x)\) is arbitrarily close to \(L\) for \(x\) sufficiently large in the negative direction.

Example 2.6.4.

Compute the limits at infinity:

  1. \(\displaystyle \displaystyle\lim_{x\rightarrow -\infty} 2 + \frac{3}{x^2}\)

  2. \(\displaystyle\lim_{x\rightarrow \infty} \frac{-2+\frac{3}{x^2}}{5-\frac{1}{x}}\text{;}\) how about \(-\infty\text{?}\)

  3. \(\displaystyle \displaystyle\lim_{x\rightarrow -\infty} \frac{\cos 5x}{-3x}\)

  4. \(\displaystyle \displaystyle\lim_{x\rightarrow \infty} \sin\left( \frac{1}{x} \right)\)

Answer.

  1. \(\displaystyle 2\)

  2. \(\displaystyle \frac{-2}{5}\)

  3. \(0\text{;}\) use \(-1 \leq \cos 5x \leq 1\) and the sandwich theorem.

  4. \(\displaystyle 0\)

Subsection 2.6.1 Rational Functions: degree of denominator \(\geq\) degree of numerator

To find \(\displaystyle\lim_{x\rightarrow\pm\infty} \frac{f(x)}{g(x)}\) for \(\deg(g) \geq \deg(f)\text{:}\) divide each term by \(x^{\deg(g)}\text{,}\) the variable raised to the degree of the denominator power.

Example 2.6.5.

  1. Let \(\displaystyle f(x)=\frac{2x^3+5x-3}{5x^3-4x^2+x}\text{;}\) find both infinite limits \(\displaystyle\lim_{x\rightarrow\pm\infty} f(x)\text{.}\)

  2. Let \(\displaystyle g(x)=\frac{3x+5}{2x^2+5x-3}\text{;}\) find \(\displaystyle\lim_{x\rightarrow\infty} g(x)\)

  3. Let \(\displaystyle h(x)=\frac{-2x^3+3\sin 5x}{4|x|^3+x^2}\text{;}\) find both \(\displaystyle\lim_{x\rightarrow\infty} h(x)\) and \(\displaystyle\lim_{x\rightarrow -\infty} h(x)\text{.}\)

Answer.

  1. \(\displaystyle \frac25\)

  2. \(\displaystyle 0\)

  3. \(\displaystyle\lim_{x\rightarrow\infty} h(x) = \frac{-1}{2}\) and \(\displaystyle\lim_{x\rightarrow -\infty} h(x) = \frac{1}{2}\)

Definition 2.6.6. Horizontal asymptote.

A line \(y=b\) is a horizontal asymptote of \(y=f(x)\) if either \(\displaystyle\lim_{x\rightarrow\infty}f(x)=b\) or \(\displaystyle\lim_{x\rightarrow -\infty}f(x)=b\text{.}\)

So, e.g., \(y=\frac{2}{3}\) is a horizontal asymptote for \(y=f(x)\text{,}\) \(y=0\) is a horizontal asymptote for \(y=g(x)\text{,}\) and \(y=-\frac{1}{2}\) and \(y=\frac{1}{2}\) are horizontal asymptotes for \(y=h(x)\text{.}\)

Subsection 2.6.2 Rational Functions: degree of denominator less than degree or numerator

To find \(\displaystyle\lim_{x\rightarrow\pm\infty} \frac{f(x)}{g(x)}\) for \(\deg(g) \lt \deg(f)\text{:}\) Use long division to write \(\frac{f(x)}{g(x)} = q(x)+\frac{r(x)}{g(x)}\text{.}\)

Definition 2.6.7. Oblique asymptote.

When the graph of \(y=f(x)\) becomes arbitrarily close to a non-horizontal line as \(x\) grows without bound in the positive or negative direction, that line is called an oblique or slant asymptote.

Example 2.6.8.

Let \(\displaystyle f(x)=\frac{x^2-36}{x-3}\text{.}\) Find the oblique asymptote.

Answer.

\(y=x+3\)

Note what happens when you try to divide by \(x\) or \(x^2\text{.}\)

Subsection 2.6.3 Limit Laws

As mentioned earlier, the limit laws apply, e.g., the rule for roots:

Example 2.6.9.

Compute \(\displaystyle\lim_{x\rightarrow\infty} \frac{x^3-7}{\sqrt{9x^6+5}}\text{.}\)

Answer.

\(\frac{1}{3}\text{;}\) note that the limit at \(-\infty\) is different

Infinity minus infinity can sometimes be resolved by multiplying by 1.

Example 2.6.10.

Find \(\displaystyle\lim_{x\rightarrow \infty} x-\sqrt{x^2+9}\text{.}\)

Solution.
\begin{align*} \lim_{x\rightarrow \infty} x-\sqrt{x^2+9}\amp =\lim_{x\rightarrow \infty} x-\sqrt{x^2+9} \frac{x + \sqrt{x^2+9}}{x+\sqrt{x^2+9}}\\ \amp =\lim_{x\rightarrow \infty} \frac{x^2 - (x^2+9)}{x+\sqrt{x^2+9}}\\ \amp =\lim_{x\rightarrow \infty} \frac{-9}{x+\sqrt{x^2+9}}\\ \amp =0 \end{align*}

Subsection 2.6.4 Infinite Limits

We write \(\displaystyle\lim_{x\rightarrow 0} \frac{1}{x^2}=\infty\text{,}\) but don't really mean it.

Definition 2.6.12.

We write \(\displaystyle\lim_{x\rightarrow c^+} f(x)=\infty\) to mean that for every positive \(B\text{,}\) for all \(x\) larger than but close enough to \(c\text{,}\) \(f(x)>B\text{.}\)

Definitions for the left and two-sided limits and \(-\infty\) are similar.

Note: this does not mean the limit exists!

Example 2.6.13.

Find \(\displaystyle\lim_{x\rightarrow 2^-} \frac{3}{x-2}\) and \(\displaystyle\lim_{x\rightarrow 2^+} \frac{3}{x-2}\text{.}\)

Answer.

\(-\infty\) and \(\infty\)

Example 2.6.14.

Rational functions can have a variety of behaviors near zeros of the denominator:

value

\(\displaystyle \displaystyle\lim_{x\rightarrow 2}\frac{x-2}{x^2-4}\)

0

\(\displaystyle \displaystyle\lim_{x\rightarrow 2}\frac{(x-2)^2}{x^2-4}\)

\(-\infty\)

\(\displaystyle \displaystyle\lim_{x\rightarrow 2^+}\frac{x-3}{x^2-4}\)

\(+\infty\)

\(\displaystyle \displaystyle\lim_{x\rightarrow 2^-}\frac{x-3}{x^2-4}\)

DNE

\(\displaystyle \displaystyle\lim_{x\rightarrow 2}\frac{x-3}{x^2-4}\)

\(-\infty\)

\(\displaystyle \displaystyle\lim_{x\rightarrow 2}\frac{2-x}{(x-2)^3}\)

Definition 2.6.15. Vertical asymptote.

A line \(x=a\) is a vertical asymptote of \(y=f(x)\) if either \(\displaystyle\lim_{x\rightarrow a^-} f(x) = \pm\infty\) or \(\displaystyle\lim_{x\rightarrow a^+} f(x) = \pm\infty\text{.}\)

Note that oblique asymptotes, e.g., \(\displaystyle f(x)=\frac{x^2-36}{x-3}\text{,}\) have infinite limits.

Subsection 2.6.5 Finding asymptotes and graphing

To find the asymptotes of the graph of a rational function \(y=f(x)\text{:}\)

Horizontal:

Divide by the largest power of \(x\) in the denominator; turn any \(\frac{\#}{x^n}\) into 0; if the result is a number, it is a horizontal asymptote. (Absolute values require considering \(+\infty\) and \(-\infty\) separately.)

Oblique:

If the degree of the numerator is larger than that of the denominator, use long division to write the function as \(q(x)+\frac{r(x)}{g(x)}\text{;}\) the limit of \(\displaystyle\lim_{x\rightarrow\pm\infty}\frac{r(x)}{g(x)}\) should now be 0 and \(y=q(x)\) is the oblique asymptote.

Vertical:

Identify points \(a\) where the denominator is 0 and evaluate the limits \(\displaystyle\lim_{x\rightarrow a^-} f(x)\) and \(\displaystyle\lim_{x\rightarrow a^+} f(x)\text{.}\)

Note: to fully answer graph sketching questions, there are four required elements: (i) identify the vertical asymptotes (if any); (ii) compute the left- and right-hand limits at any vertical asymptotes; (iii) Find horizontal or oblique asymptotes (if any); and (iv) sketch a graph incorporating all of the former.

Example 2.6.17.

Find the horizontal and oblique asymptotes of

\begin{equation*} f(x) = \frac{6x^3-3x^2+3x-1}{-2x^2+15x-22} \end{equation*}

and graph the function.

Answer.

oblique asymptote: \(y=-3x+21\)

vertical asymptotes: \(0=-2x^2+15x-22=(2-x)(2x-11)\text{,}\) so \(x=2\) and \(x=\frac{11}{2}\text{.}\)

limits: \(\displaystyle\lim_{x\to 2^-} f(x) = -\infty\text{,}\) \(\displaystyle\lim_{x\to 2^+} f(x) = \infty\text{,}\) \(\displaystyle\lim_{x\to 11/2^-} f(x) = \infty\text{,}\) \(\displaystyle\lim_{x\to 11/2^+} f(x) = -\infty\text{.}\)

Example 2.6.18.

Find the horizontal and vertical asymptotes of

\begin{equation*} f(x)=\frac{3-x}{x-4} \end{equation*}

and graph the function.

Answer.

horizontal asymptote: \(y=-1\)

vertical asymptote: \(x=4\text{,}\) with \(\displaystyle\lim_{x\to 4^-} f(x) = \infty\) and \(\displaystyle\lim_{x\to 4^-} f(x) = -\infty\)

graph: https://www.desmos.com/calculator/1o5lnefcz9 1 

Example 2.6.19.

Find the horizontal/oblique and vertical asymptotes of

\begin{equation*} g(x)=\frac{6x^2}{2x+1} \end{equation*}

and graph the function.

Answer.

oblique asymptote: \(y=3x-3/2\)

vertical asymptote: \(x=-1/2\text{,}\) \(\displaystyle\lim_{x\to \frac{-1}{2}^-} g(x) = -\infty\) and \(\displaystyle\lim_{x\to \frac{-1}{2}^+} g(x) = +\infty\)

graph: https://www.desmos.com/calculator/bhg4vjjxyg 2 

www.desmos.com/calculator/hbqsyvinfl
www.desmos.com/calculator/bhg4vjjxyg