Question

find the limit: lim(x->0) sqrt(x+4)-2/x

Answer 1

if it is:
\(\lim_{x\to 0}\sqrt{x+4}-\frac{2}{x}=-\infty\)
if it is:
\(\lim_{x\to 0}\frac{\sqrt{x+4}-2}{x}\) use L'Hopital's rule

Answer 2

In that second case you might also try this "trick":
\( \begin{align}
\dfrac{\sqrt{x + 4} - 2}{x} &= \dfrac{\sqrt{x + 4} - 2}{x + 4 - 4} \\
&= \dfrac{\sqrt{x + 4} - 2}{(\sqrt{x + 4} + 2)(\sqrt{x + 4} - 2)} \\
\end{align}\)
Which allows you to cancel out the problem value and use direct substitution. :)

Answer 3

Rationalize the numerator.