question evaluate the limit: $lim_{x\rightarrow12}\frac{sqrt{x + 4}-4}{x - 12}$ examples
question evaluate the limit: $lim_{x\rightarrow12}\frac{sqrt{x + 4}-4}{x - 12}$ examples
Answer
Answer:
$\frac{1}{8}$
Explanation:
Step1: Rationalize the numerator
Multiply the fraction by $\frac{\sqrt{x + 4}+4}{\sqrt{x + 4}+4}$. [ \begin{align*} &\lim_{x\rightarrow12}\frac{\sqrt{x + 4}-4}{x - 12}\times\frac{\sqrt{x + 4}+4}{\sqrt{x + 4}+4}\ =&\lim_{x\rightarrow12}\frac{(\sqrt{x + 4})^2-4^2}{(x - 12)(\sqrt{x + 4}+4)}\ =&\lim_{x\rightarrow12}\frac{x + 4-16}{(x - 12)(\sqrt{x + 4}+4)}\ =&\lim_{x\rightarrow12}\frac{x - 12}{(x - 12)(\sqrt{x + 4}+4)} \end{align*} ]
Step2: Simplify the fraction
Cancel out the common factor $(x - 12)$ in the numerator and denominator. [ \begin{align*} &\lim_{x\rightarrow12}\frac{x - 12}{(x - 12)(\sqrt{x + 4}+4)}\ =&\lim_{x\rightarrow12}\frac{1}{\sqrt{x + 4}+4} \end{align*} ]
Step3: Substitute the limit value
Substitute $x = 12$ into the simplified function. [ \begin{align*} &\frac{1}{\sqrt{12 + 4}+4}\ =&\frac{1}{\sqrt{16}+4}\ =&\frac{1}{4 + 4}\ =&\frac{1}{8} \end{align*} ]