# How to sketch a graph of this function and determine if f is continuous at x = 3?

Jan 29, 2017

There are infinitely many examples of functions that have $f \left(3\right) = 1$ and ${\lim}_{x \rightarrow 3} f \left(x\right) = 1$.

#### Explanation:

The graph is not needed to see that $f$ is continuous at $3$. The definition of "continuous at $3$" is ${\lim}_{x \rightarrow 3} f \left(x\right) = f \left(3\right)$. And we are told that this is true.

The graph of $f$ must include the point $\left(3 , 1\right)$ and for values of $x$ near $3$, the corresponding $y$ values must be close to 1`.

Another way to say this is: near $x = 3$, the graph must be connected (on both sides) to $\left(3 , 1\right)$.

The first graph does not have $f \left(3\right) = 1$ (There is an open hole at $\left(3 , 1\right)$)
It also does not have ${\lim}_{x \rightarrow 3} f \left(3\right) = 1$ (The limit is $1$ from the left and $- 1$ from the right.
The second graph has $f \left(3\right) = 1$, but again, it has limit $1$ only from the left.
The third graph does have ${\lim}_{x \rightarrow 3} f \left(x\right) = 1$, but the hole at $\left(3 , 1\right)$ indicates that $f \left(3\right) \ne 1$. (In fact #f(3) does not exist on the third graph.)