Question:
If $f(x)$ is continuous at $x=a$ and $\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=k$, then $k$ is equal to____________
Solution:
It is given that, f(x) is continuous at x = a.
$\therefore f(a)=\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)$ ....(1)
Also,
$\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=k$ .....(2)
From (1) and (2), we have
$f(a)=k$
Thus, the value of k is f(a).
If $f(x)$ is continuous at $x=a$ and $\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=k$, then $k$ is equal to $f(a)$