2.2 The limit of a function  (Page 7/12)

Key concepts

• A table of values or graph may be used to estimate a limit.
• If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
• If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
• We may use limits to describe infinite behavior of a function at a point.

Key equations

• Intuitive Definition of the Limit
  $\underset{x\to a}{\text{lim}}f\left(x\right)=L$
• Two Important Limits
  $\underset{x\to a}{\text{lim}}x=a\underset{x\to a}{\text{lim}}c=c$
• One-Sided Limits
  $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=L\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=L$
• Infinite Limits from the Left
  $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{+}\infty \underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty$
• Infinite Limits from the Right
  $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{+}\infty \underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty$
• Two-Sided Infinite Limits
  $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty :\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{+}\infty$ and $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{+}\infty$
  $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{−}\infty :\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty$ and $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty$

For the following exercises, consider the function $f\left(x\right)=\frac{x^2-1}{\left|x-1\right|}.$

For the following exercises, consider the function $f\left(x\right)={\left(1+x\right)}^{1\text{/}x}.$

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.