Question

Evaluate the following limit Show all process step by step

Answer 1

AnswEr :

Given,

$\sf \: lim. \: \: \: \: \: \: \: \dfrac{1 - \tan(x) }{x - \frac{\pi}{4} } \\ \sf{x \longrightarrow \: \frac{\pi}{4} }$

At π/4,the above function tends to 0/0 form which is indeterminate.

Using L'Hospital's Rule,

$\implies \: \sf \: lim. \: \: \: \: \: \: \: \dfrac{ \dfrac{d(1 - \tan(x)) }{dx}}{ \dfrac{d(x - \: \frac{\pi}{4})}{dx} } \\ \: \: \: \: \: \: \: \: \sf{x \longrightarrow \: \frac{\pi}{4} }$

Therefore,

$\implies \: \sf \: lim. \: \: \: \: \: \: \: \dfrac{0 - {sec}^{2}x }{1 - 0} \\ \: \: \: \: \: \: \: \: \sf{x \longrightarrow \: \frac{\pi}{4} }$

Substituting the limit,we get :

$\implies \: \sf \: - sec {}^{2} ( \dfrac{\pi}{4} ) \\ \\ \implies \sf - ( \sqrt{2} ) {}^{2} \\ \\ \implies \sf - 2$

Thus,the above function becomes - 2 for x tending to π/4