Question

find the limit. lim h→0 (√(5h + 1) - 1)/h select the correct choice below and, if necessary, fill in the answer box to complete your choice a. lim h→0 (√(5h + 1) - 1)/h = (type an integer or a simplified fraction.) b. the limit does not exist

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{5h + 1}+1}{\sqrt{5h + 1}+1}$.
\[

$$\begin{align*} \lim_{h ightarrow0}\frac{\sqrt{5h + 1}-1}{h}&=\lim_{h ightarrow0}\frac{(\sqrt{5h + 1}-1)(\sqrt{5h + 1}+1)}{h(\sqrt{5h + 1}+1)}\\ \end{align*}$$

\]
Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, we get $\lim_{h
ightarrow0}\frac{(5h + 1)-1}{h(\sqrt{5h + 1}+1)}$.

Step2: Simplify the numerator

Simplify the numerator: $(5h + 1)-1 = 5h$. So the limit becomes $\lim_{h
ightarrow0}\frac{5h}{h(\sqrt{5h + 1}+1)}$.

Step3: Cancel out the common factor

Cancel out the common factor $h$ (since $h
eq0$ as we are taking the limit as $h$ approaches 0, not setting $h = 0$). We have $\lim_{h
ightarrow0}\frac{5}{\sqrt{5h + 1}+1}$.

Step4: Evaluate the limit

Substitute $h = 0$ into the expression $\frac{5}{\sqrt{5h + 1}+1}$. We get $\frac{5}{\sqrt{5\times0+1}+1}=\frac{5}{\sqrt{1}+1}=\frac{5}{2}$.

Answer:

A. $\frac{5}{2}$