12.1 Finding limits: numerical and graphical approaches  (Page 3/10)

Understanding two-sided limits

In the previous example, the left-hand limit and right-hand limit as $x$ approaches $a$ are equal. If the left- and right-hand limits are equal, we say that the function $f(x)$ has a two-sided limit    as $x$ approaches $a.$ More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

Finding a limit using a graph

To visually determine if a limit exists as $x$ approaches $a,$ we observe the graph of the function when $x$ is very near to $x=a.$ In [link] we observe the behavior of the graph on both sides of $a.$

To determine if a left-hand limit exists, we observe the branch of the graph to the left of $x=a,$ but near $x=a.$ This is where $x<a.$ We see that the outputs are getting close to some real number $L$ so there is a left-hand limit.

To determine if a right-hand limit exists, observe the branch of the graph to the right of $x=a,$ but near $x=a.$ This is where $x>a.$ We see that the outputs are getting close to some real number $L,$ so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in [link] , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function $f\left(x\right)$ when the input value $x$ is equal to $a.$ The coordinate pair of the point would be $\left(a,f\left(a\right)\right).$ If such a point exists, then $f\left(a\right)$ has a value. If the point does not exist, as in [link] , then we say that $f\left(a\right)$ does not exist.