Estimating a limit numerically near a point where the function is undefined

Evaluating $f(x)$ at values like $x = \pm 0.1, \pm 0.01, \pm 0.001$ gives outputs that all round to approximately $0.25$ regardless of whether $x$ approaches $0$ from the left or the right. For instance, $f(0.01) = \dfrac{\sqrt{4.01}-2}{0.01} \approx 0.24998$ and $f(-0.01) = \dfrac{\sqrt{3.99}-2}{-0.01} \approx 0.25002$.

Since the values from both sides settle toward the same number as $x$ gets closer to $0$, we conclude the two-sided limit exists and equals that common value. Note that $f(0)$ itself is not defined (it's $0/0$), but this doesn't matter for the limit — only the behavior of nearby values matters, and here $\lim_{x\to0} f(x) = 0.25 = \tfrac{1}{4}$.