How do you use a graph to show that the limit does not exist?

Sep 27, 2014

Remember that limits represent the tendency of a function, so limits do not exist if we cannot determine the tendency of the function to a single point. Graphically, limits do not exist when:

1. there is a jump discontinuity
(Left-Hand Limit $\ne$ Right-Hand Limit)
The limit does not exist at $x = 1$ in the graph below.

1. there is a vertical asymptote
(Infinit Limit)
(Caution: When you have infinite limits, limits do not exist.)
The limit at $x = 2$ does not exist in the graph below.

1. there is a violent oscillation
(e.g., $\sin \left(\frac{1}{x}\right)$ at $x = 0$, shown below)

I hope that this was helpful.