Answer
$$\frac{2}{\sqrt{5}} $$
Work Step by Step
\begin{aligned}
\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+4 h+5}-\sqrt{5}}{h} &=\lim _{h \rightarrow 0^{+}}\left(\frac{\sqrt{h^{2}+4 h+5}-\sqrt{5}}{h}\right)\left(\frac{\sqrt{h^{2}+4 h+5}+\sqrt{5}}{\sqrt{h^{2}+4 h+5}+\sqrt{5}}\right)\\
&=\lim _{h \rightarrow 0^{+}} \frac{\left(h^{2}+4 h+5\right)-5}{h(\sqrt{h^{2}+4 h+5}+\sqrt{5})} \\
&=\lim _{h \rightarrow 0^{+}} \frac{h(h+4)}{h(\sqrt{h^{2}+4 h+5}+\sqrt{5})}\\
&=\frac{0+4}{\sqrt{5}+\sqrt{5}}\\
&=\frac{2}{\sqrt{5}} \end{aligned}
