$\lim_{x \to a} f(x) = L$, read aloud: the limit of f of x, as x approaches a, is L. Here $f$ is the function, $a$ is the input being approached, and $L$ is the number the outputs close in on as the inputs get close to $a$. Close to, never equal to: the limit asks what $f$ is DOING near $a$, and the value $f(a)$ itself gets no vote. It might be the same number, it might be a different number, it might not exist. The limit does not care.

Three pictures to keep. A smooth curve through $(a, L)$: the limit is $L$ and $f(a)$ happens to agree. The same curve with a hole at $x = a$: the limit is still $L$, because the neighbours still point there; $f(a)$ is simply undefined. The same curve with a hole at $x = a$ and a solid dot floating at some other height: the limit is still $L$ and $f(a)$ is the dot. One limit, three different $f(a)$. The exam's favourite trap is a dot drawn somewhere obvious and a question that never asked about it.