The statement in the image reads:

"When lim f(x) exists, it always equals f(a). State whether this statement is true or false."

This statement is false.

Explanation:

For the limit $\lim_{{x \to a}} f(x)$ to exist, the function $f(x)$ must approach a particular value as $x$ gets closer to $a$. However, this does not necessarily mean that the value of the function at $x = a$, i.e., $f(a)$, is the same as the limit.

• In order for $\lim_{{x \to a}} f(x) = f(a)$, the function must be continuous at $x = a$, meaning that:
  • $\lim_{{x \to a}} f(x)$ exists.
  • $f(a)$ is defined.
  • $\lim_{{x \to a}} f(x) = f(a)$.

However, there are cases where $\lim_{{x \to a}} f(x)$ exists, but $f(a)$ is either undefined or not equal to the limit. This situation occurs in discontinuous functions.

Example of a discontinuous function:

Consider the function: