evaluate the limit: lim(x→0) (√(5x + 49) - 7)/x

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the denominator
Multiply the numerator and denominator by $\sqrt{5x + 49}+7$.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{\sqrt{5x + 49}-7}{x}\times\frac{\sqrt{5x + 49}+7}{\sqrt{5x + 49}+7}\
=&\lim_{x
ightarrow0}\frac{(5x + 49)-49}{x(\sqrt{5x + 49}+7)}\
=&\lim_{x
ightarrow0}\frac{5x}{x(\sqrt{5x + 49}+7)}
\end{align*}
$$
## Step2: Simplify the expression
Cancel out the common - factor $x$ in the numerator and denominator.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{5x}{x(\sqrt{5x + 49}+7)}\
=&\lim_{x
ightarrow0}\frac{5}{\sqrt{5x + 49}+7}
\end{align*}
$$
## Step3: Evaluate the limit
Substitute $x = 0$ into the simplified expression.
$$
\begin{align*}
&\frac{5}{\sqrt{5\times0 + 49}+7}\
=&\frac{5}{\sqrt{49}+7}\
=&\frac{5}{7 + 7}\
=&\frac{5}{14}
\end{align*}
$$
# Answer:
$\frac{5}{14}$

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Question

Explanation:

Step1: Rationalize the denominator

Step2: Simplify the expression

Step3: Evaluate the limit

Answer: