Infinite Limits

Consider the following function:   \(f(x)=\large{\frac{1}{x+3}}\normalsize{+2}\).   At \(x=-3\) something strange is going on.   The graph and a table are shown below:

graph table

From either analyzing the graph or looking at the table, as \(x\rightarrow-3^-\), the function values are getting more and more negative, hence smaller.   This behavior is called "approaching negative infinity", meaning the limit Does Not Exist and is expressed as \(\lim_{x\rightarrow-3^-}f(x)=-\infty\).

Likewise, as \(x\rightarrow-3^+\), the function values grow larger and larger, "approach positive infinity", so the limit does not exist and is expressed as \(\lim_{x\rightarrow-3^+}f(x)=\infty\).   This type of function behavior is referred to as an infinite limit.   This leads to the following definitions:

One-Sided Infinite Limits

Suppose \(f\) is defined for all \(x\) near \(a\) with \(x>a\).   If \(f(x)\) grows arbitrarily large for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a^+}f(x)=\infty\).

Suppose \(f\) is defined for all \(x\) near \(a\) with \(x< a\).   If \(f(x)\) grows arbitrarily large for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a^-}f(x)=\infty\).

One-Sided Infinite Limits

Suppose \(f\) is defined for all \(x\) near \(a\) with \(x>a\).   If \(f(x)\) grows arbitrarily small for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a^+}f(x)=-\infty\).

Suppose \(f\) is defined for all \(x\) near \(a\) with \(x< a\).   If \(f(x)\) grows arbitrarily small for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a^-}f(x)=-\infty\).

Two-Sided Infinite Limits

Suppose \(f\) is defined for all \(x\) near \(a\).   If \(f(x)\) grows arbitrarily large for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a}f(x)=\infty\).

Suppose \(f\) is defined for all \(x\) near \(a\).   If \(f(x)\) grows arbitrarily small for all \(x\) sufficiently close to \(a\), then \(\lim_{x\rightarrow a}f(x)=-\infty\).


The above definitions, \(\lim_x{\rightarrow a}f(x)=\pm\infty,\;\lim_x{\rightarrow a^-}f(x)=\pm\infty\) and \(\lim_x{\rightarrow a^+}f(x)=\pm\infty\), lead to a graphical interpretation about the line \(x=a\).   This tending to infinity as \(x\rightarrow a\) defines a vertical asymptote at \(x=a\)..

Limits Analytically

To anaylze limits, first consider a very important result. What happens as \(x\rightarrow0\) in the function \(f(x)=\large{\frac{1}{x}}\)?

reciprocal function

The table shows that as \(x\rightarrow0^+\), the function values seem to be growing without bound.

This leads to the conclusion:   \(\lim_{x\rightarrow0^+}\left(\large{\frac{1}{x}}\right)=\infty\)

Looking at \(x\rightarrow0^-\), recall this means \(x< 0\), so the \(x\)-values would be negative, which also makes the function values negative.

This leads to the conclusion:   \(\lim_{x\rightarrow0^-}\left(\large{\frac{1}{x}}\right)=-\infty\).

This result is used to help analyze limits analytically.

Example

Analyze the limit:   \(\lim_{x\rightarrow0^-}\left(\large{\frac{x-5}{x}}\right)\).

Direct substitution shows the numerator approaches \(-5\).

Since \(x\rightarrow0^-\), the \(x\)-values are negative and approaching zero.		 $$\lim_{x\rightarrow0^-}\left(\frac{x-5}{x}\right)\implies \lim_{x\rightarrow0^-}\frac{\overset{\color{blue}{\text{approaches }-5}}{x-5}}{\underset{\color{green}{\text{approaches 0 negatively}}}{x}}$$
Both the numerator and denominator approach negative values, this means overall the function values approach positive infinity. $$\lim_{x\rightarrow0^-}\left(\frac{x-5}{x}\right)=\infty\;\color{green}{\checkmark}$$
Example

Analyze the limit:   \(\lim_{x\rightarrow2^+}\left(\large{\frac{x^2-4x+3}{x^2-4x+4}}\right)\).

Simplify the function first, see if cancellation occurs. \(\large{\frac{x^2-4x+3}{x^2-4x+4}}\normalsize{\implies}\large{\frac{(x-3)(x-1)}{(x-2)^2}}\)
No cancellation, plug in \(x=2\): \(\large{\frac{(2-3)(2-1)}{(2-2)^2}}\normalsize{\implies}\large{\frac{(-1)(1)}{0}}\)
As \(x\rightarrow2^+\): \(\lim_{x\rightarrow2^+}\left(\large{\frac{\overset{\color{blue}{\text{approaches -1}}}{x^2-4x+3}}{\underset{\color{green}{\text{approaches 0 positively}}}{x^2-4x+4}}}\right)=-\infty\;\color{green}{\checkmark}\)
Example

Analyze the limit:   \(\lim_{x\rightarrow-\pi^-}\tan\left(x+\frac{\pi}{2}\right)\).