Question

find the limit.
lim (sqrt(h^2 + 2h + 13) - sqrt(13))/h
h->0+
lim (sqrt(h^2 + 2h + 13) - sqrt(13))/h=
h->0+

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{h^{2}+2h + 13}+\sqrt{13}}{\sqrt{h^{2}+2h + 13}+\sqrt{13}}$.
\[

$$\begin{align*} &\lim_{h ightarrow0^{+}}\frac{\sqrt{h^{2}+2h + 13}-\sqrt{13}}{h}\times\frac{\sqrt{h^{2}+2h + 13}+\sqrt{13}}{\sqrt{h^{2}+2h + 13}+\sqrt{13}}\\ =&\lim_{h ightarrow0^{+}}\frac{(h^{2}+2h + 13)-13}{h(\sqrt{h^{2}+2h + 13}+\sqrt{13})}\\ =&\lim_{h ightarrow0^{+}}\frac{h^{2}+2h}{h(\sqrt{h^{2}+2h + 13}+\sqrt{13})} \end{align*}$$

\]

Step2: Simplify the fraction

Cancel out the common - factor $h$ in the numerator and denominator.
\[

$$\begin{align*} &\lim_{h ightarrow0^{+}}\frac{h^{2}+2h}{h(\sqrt{h^{2}+2h + 13}+\sqrt{13})}\\ =&\lim_{h ightarrow0^{+}}\frac{h(h + 2)}{h(\sqrt{h^{2}+2h + 13}+\sqrt{13})}\\ =&\lim_{h ightarrow0^{+}}\frac{h + 2}{\sqrt{h^{2}+2h + 13}+\sqrt{13}} \end{align*}$$

\]

Step3: Evaluate the limit

Substitute $h = 0$ into the simplified function.
\[

$$\begin{align*} &\frac{0 + 2}{\sqrt{0^{2}+2\times0+13}+\sqrt{13}}\\ =&\frac{2}{2\sqrt{13}}=\frac{1}{\sqrt{13}} \end{align*}$$

\]

Answer:

$\frac{1}{\sqrt{13}}$