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

Steps to Solve

1. Understanding the limit concept The statement involves the limit of a function as $x$ approaches $a$, expressed as $\lim_{{x \to a}} f(x)$. For the limit to exist and equal to $f(a)$, $f(a)$ must be defined and continuous at $a$.

2. Identifying the conditions for limit equality For $\lim_{{x \to a}} f(x) = f(a)$ to be true, the following must hold:

   • $f(a)$ is defined (i.e., $f(a)$ exists).
   • The function $f(x)$ must approach the same value from both the left and right as $x$ approaches $a$.

3. Analyzing the answer options Option A suggests that evaluating $f(x)$ at $x = a$ ensures the limit equals $f(a)$; however, this only holds if $f(a)$ is defined.

   Option B states that the limit cannot exist if $f(a)$ is not defined. This is true but doesn't address the limit existing even when $f(a)$ is defined.

   Option C correctly states that the limit depends on values near $a$, but does not directly address limit equality with $f(a)$.

   Option D emphasizes that if $x = a$, then $f(x) = f(a)$; this can be misleading unless $f(a)$ is defined.

4. Conclusion based on definitions The statement is false in general because the limit does not always equal the function value at that point unless specified conditions are met.