Question

lim(x tends to p) (x^2-p^2)/tan(x-p)​

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf \: \lim_{x \to p} \: \frac{ {x}^{2} - {p}^{2} }{ \tan (x - p)}$

EVALUATION

Here the given limit is

$\displaystyle \sf \: \lim_{x \to p} \: \frac{ {x}^{2} - {p}^{2} }{ \tan (x - p)}$

We simplify it as below

$\displaystyle \sf \: \lim_{x \to p} \: \frac{ {x}^{2} - {p}^{2} }{ \tan (x - p)} \: \: \: \: \bigg( \frac{0}{0} \: form \bigg)$

$\displaystyle \sf \: = \lim_{x \to p} \: \frac{ 2x - 0 }{ 1. { \sec}^{2} (x - p)}$

$\displaystyle \sf \: = \lim_{x \to p} \: \frac{ 2x }{ { \sec}^{2} (x - p)}$

$\displaystyle \sf \: = \: \frac{ 2p }{ { \sec}^{2} (p - p)}$

$\displaystyle \sf \: = \: \frac{ 2p }{ { \sec}^{2} 0 }$

$\displaystyle \sf \: = \: 2p$

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