# Thread: When a limit is equal to a number...

1. ## When a limit is equal to a number...

what is that number ... like if we had a limit that was equal to 3 is 3 the point where the graph is undefined??

2. Originally Posted by yess
what is that number ... like if we had a limit that was equal to 3 is 3 the point where the graph is undefined??
I find this to be a very strange question.. incoherent really. Maybe you should go back and review the definition of limit?

Graphical explanation may help. On the left is graph of f(x)=x. On right is graph of

$\displaystyle f(x)=\begin{cases} x&,\ x\ne2\\ 4&,\ x=2\end{cases}$

On the left we have $\displaystyle \displaystyle \lim_{x\to2}f(x)=2$ and $\displaystyle \,f(2)=2$. On the right we have $\displaystyle \displaystyle \lim_{x\to2}f(x)=2$ and $\displaystyle \,f(2)=4$.

3. Originally Posted by undefined
I find this to be a very strange question.. incoherent really. Maybe you should go back and review the definition of limit?

Graphical explanation may help. On the left is graph of f(x)=x. On right is graph of

$\displaystyle f(x)=\begin{cases} x&,\ x\ne2\\ 4&,\ x=2\end{cases}$

On the left we have $\displaystyle \displaystyle \lim_{x\to2}f(x)=2$ and $\displaystyle \,f(2)=2$. On the right we have $\displaystyle \displaystyle \lim_{x\to2}f(x)=2$ and $\displaystyle \,f(2)=4$.
so like when we say "the limit is 4" that means the graph is discontinuous at x=4??

4. Originally Posted by yess
so like when we say "the limit is 4" that means the graph is discontinuous at x=4??
No.

Referring to the graphs in my first post: for the left one, f(x) is continuous at x=2 because

1) the limit of f(x) as x approaches two exists
2) that limit is equal to the value of f at that point.

In symbols, $\displaystyle \displaystyle \lim_{x\to2}f(x)=f(2)$.

The graph on the right is discontinuous at x=2 because, although the limit exists, it does not equal the value of the function at that point.

Informally continuity is often described as "it's continuous if you can draw it without lifting your pen," and applying that to the graph on the right, it is not continuous at x=2 because the function "jumps" and you have to lift your pen there.