Question

Evaluate the limit:
$\lim_{n \to 16} \frac{\sqrt{x} -4}{x-16}$

Answer 1

Appropriate Question

Evaluate the limit

$\rm :\longmapsto\:\displaystyle\lim_{x \to 16} \frac{ \sqrt{x} - 4 }{x - 16}$

$\red{\large\underline{\sf{Solution-}}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{x \to 16} \frac{ \sqrt{x} - 4 }{x - 16}$

If we substitute directly x = 16, we get

$\rm \:  =  \: \dfrac{ \sqrt{16} - 4 }{16 - 16}$

$\rm \:  =  \: \dfrac{ 4 - 4 }{0}$

$\rm \:  =  \: \dfrac{0}{0}$

which is indeterminant form,

So,

$\rm :\longmapsto\:\displaystyle\lim_{x \to 16} \frac{ \sqrt{x} - 4 }{x - 16}$

To evaluate this limit, we use Method of Substitution

So, we substitute

$\red{\rm :\longmapsto\:x = {y}^{2}, \: as \: x \: \to \: 16 \: \: y \: \to \: 4}$

So, above expression can be rewritten as

$\rm \:  =  \: \displaystyle\lim_{y \to 4} \frac{y - 4}{ {y}^{2} - 16}$

can be rewritten as

$\rm \:  =  \: \displaystyle\lim_{y \to 4} \frac{y - 4}{ {y}^{2} - {4}^{2} }$

$\rm \:  =  \: \displaystyle\lim_{y \to 4} \frac{y - 4}{ (y - 4)(y + 4) }$

$\rm \:  =  \: \displaystyle\lim_{y \to 4} \frac{1}{y + 4}$

$\rm \:  =  \: \dfrac{1}{4 + 4}$

$\rm \:  =  \: \dfrac{1}{8}$

Hence,

$\rm \implies\:\:\boxed{ \tt{ \: \displaystyle\lim_{x \to 16} \frac{ \sqrt{x} - 4 }{x - 16} = \frac{1}{8} \: }}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional Information

$\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \frac{sinx}{x} = 1 \: }}$

$\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \frac{tanx}{x} = 1 \: }}$

$\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \frac{log(1 + x)}{x} = 1 \: }}$

$\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \frac{ {e}^{x} - 1}{x} = 1 \: }}$

$\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \frac{ {a}^{x} - 1}{x} = loga \: }}$