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+

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+

Accepted Answer

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

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Simplify the fraction

Step3: Evaluate the limit

Answer: