A function is continuous at a point if there it is defined and there are no breaks at the point. These functions are continuous everywhere.

This function is not continuous at x = 1.

This function is not continuous at x = -2

A limit is a number (y-value) that a function approaches, but doesn’t necessarily equal.

 

 

In general if a function is defined and has no unusual characteristics, the functional value is equal to the limit.

The time when we most often care about limits is when functions are undefined or have breaks (discontinuities) in them.

The limit is the value you get close to, not necessarily the y-value the function equals at a given point.

If you are given a function formula … 1. Factor 2. Cancel 3. Plug in

If you can cancel and find a limit, this is called a removable discontinuity. This basically means there’s a hole in the graph of a function.

Sometimes a limit doesn’t exist Sometimes a limit doesn’t exist. This is because the y-value doesn’t go to one specific value as x approaches a number.

For instance, it could approach two different numbers:

The y-value could also become infinitely large.

Here’s a combination of both problems.

Yet another problem that means a limit doesn’t exist is oscillating behavior.