Question

The graph of a function f is shown above. If lim f (x) exists, and f is not continuous at2= b, then b=

Answer 1

Recall that

$\lim_{x\to b}f(x)$

exists if and only if:

$\lim_{x\to b^-}f(x)=\lim_{x\to b+}f(x).$

Now, from the given graph we get that:

$\begin{gathered} \lim_{x\to0^+}f(x)=2, \\ \lim_{x\to0^-}f(x)=2, \end{gathered}$

Then:

$\lim_{x\to0^-}f(x)=\lim_{x\to0^+}f(x).$

Therefore:

$\lim_{x\to0}f(x)$

exists.

Now, from the given graph we get that:

$f(0)<2.$

Therefore f(x) is not continuous at x=0.

Answer: b=0.