The value of mathop {lim }limits_{x to 0} fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) We know that $\frac{\pi}{4} = 	an^{-1}(1)$.Let $L = \mathop {\lim }\limits_{x 	o 0} \frac{1}{x}\left[ {{{	an }^{ - 1}}\left( {\frac{{x + 1}}{{2

Vedclass · Accepted Answer

$-\frac{1}{2}$

Vedclass · Answer

(B) We know that $\frac{\pi}{4} = 	an^{-1}(1)$.Let $L = \mathop {\lim }\limits_{x 	o 0} \frac{1}{x}\left[ {{{	an }^{ - 1}}\left( {\frac{{x + 1}}{{2x + 1}}} ight) - {{	an }^{ - 1}}(1)} ight]$.Using the formula $	an^{-1}(A) - 	an^{-1}(B) = 	an^{-1}\left(\frac{A-B}{1+AB}ight)$,we get:$L = \mathop {\lim }\limits_{x 	o 0} \frac{1}{x} 	an^{-1}\left( \frac{\frac{x+1}{2x+1} - 1}{1 + \frac{x+1}{2x+1}} ight)$$L = \mathop {\lim }\limits_{x 	o 0} \frac{1}{x} 	an^{-1}\left( \frac{x+1 - 2x - 1}{2x+1 + x+1} ight) = \mathop {\lim }\limits_{x 	o 0} \frac{1}{x} 	an^{-1}\left( \frac{-x}{3x+2} ight)$Using $\mathop {\lim }\limits_{	heta 	o 0} \frac{	an^{-1}(	heta)}{	heta} = 1$,we multiply and divide by $\frac{-x}{3x+2}$:$L = \mathop {\lim }\limits_{x 	o 0} \left( \frac{	an^{-1}\left( \frac{-x}{3x+2} ight)}{\frac{-x}{3x+2}} ight) 	imes \left( \frac{1}{x} \cdot \frac{-x}{3x+2} ight)$$L = 1 	imes \mathop {\lim }\limits_{x 	o 0} \left( \frac{-1}{3x+2} ight) = -\frac{1}{2}$.

The value of $\mathop {\lim }\limits_{x \to 0} \frac{1}{x}\left[ {{{\tan }^{ - 1}}\left( {\frac{{x + 1}}{{2x + 1}}} \right) - \frac{\pi }{4}} \right]$ is

Similar Questions

The value of $\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {1 + \sqrt {2 + x} } - \sqrt 3 }}{{x - 2}}$ is

If $n$ is an integer,then $\mathop {\lim }\limits_{x \to n + 0} (x - [x]) = $

If $\Delta(x) = \begin{vmatrix} e^x & -1 \\ \sin x - 1 & 1 \end{vmatrix}$,then $\lim_{x \rightarrow 0} \frac{\Delta(x)}{x} = $

The function $f(x) = \lim_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1}$ is identical to which of the following functions?

If the value of $\lim _{x \rightarrow 0}(2-\cos x \sqrt{\cos 2 x})^{\left(\frac{x+2}{x^{2}}\right)}$ is equal to $e^{a}$,then $a$ is equal to $.....$

Vedclass Test Series

Exam Paper Generator

Online Exam Module