Q.If $f(x) = \dfrac{24^x-8^x-3^x+1}{12^x-4^x-3^x+1}$, for $x \ne 0$, $= k$, for $x = 0$, is continuous at $x = 0$, find $k$.

A function $f(x)$ is continuous at a point $x=a$ when three separate requirements all hold together: the function must actually be defined at $a$ (and on a whole open interval around it), the limit $\lim_{x\to a} f(x)$ must exist as a single finite number, and that limiting value must be exactly equal to $f(a)$ itself. If any one of these three fails — the value is missing, the two-sided limit does not exist, or the limit disagrees with the assigned value — the function is discontinuous at that point. An equivalent way to state the third condition is $\lim_{h\to0}[f(a+h)-f(a)]=0$, which says that as the input is nudged away from $a$ by a shrinking amount $h$, the resulting change in output must also shrink to zero, ruling out any sudden jump rig …