Question

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

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:

$\frac{5}{14}$