mathop {lim }limits_{theta to pi /2} (sec the | vedclass

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

Question

(A) We need to evaluate $\mathop {\lim }\limits_{	heta 	o \pi /2} (\sec 	heta - 	an 	heta )$.Substituting $\sec 	heta = \frac{1}{\cos 	heta}$ a

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) We need to evaluate $\mathop {\lim }\limits_{	heta 	o \pi /2} (\sec 	heta - 	an 	heta )$.Substituting $\sec 	heta = \frac{1}{\cos 	heta}$ and $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$,we get:$\mathop {\lim }\limits_{	heta 	o \pi /2} \left( \frac{1 - \sin 	heta}{\cos 	heta} ight)$.As $	heta 	o \pi /2$,this is a $\frac{0}{0}$ indeterminate form.Using the identity $1 - \sin 	heta = \left( \cos \frac{	heta}{2} - \sin \frac{	heta}{2} ight)^2$ and $\cos 	heta = \cos^2 \frac{	heta}{2} - \sin^2 \frac{	heta}{2} = (\cos \frac{	heta}{2} - \sin \frac{	heta}{2})(\cos \frac{	heta}{2} + \sin \frac{	heta}{2})$:$\mathop {\lim }\limits_{	heta 	o \pi /2} \frac{(\cos \frac{	heta}{2} - \sin \frac{	heta}{2})^2}{(\cos \frac{	heta}{2} - \sin \frac{	heta}{2})(\cos \frac{	heta}{2} + \sin \frac{	heta}{2})} = \mathop {\lim }\limits_{	heta 	o \pi /2} \frac{\cos \frac{	heta}{2} - \sin \frac{	heta}{2}}{\cos \frac{	heta}{2} + \sin \frac{	heta}{2}}$.Substituting $	heta = \pi / 2$,we get $\frac{\cos(\pi/4) - \sin(\pi/4)}{\cos(\pi/4) + \sin(\pi/4)} = \frac{1/\sqrt{2} - 1/\sqrt{2}}{1/\sqrt{2} + 1/\sqrt{2}} = \frac{0}{\sqrt{2}} = 0$.

$\mathop {\lim }\limits_{\theta \to \pi /2} (\sec \theta - \tan \theta ) = $

Similar Questions

$\mathop {\lim }\limits_{x \to 1} \frac{{1 + \cos \pi x}}{{{{\tan }^2}\pi x}}$ is equal to

$\lim _{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1} = $

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^3-x^2-x-2}{2 x^3-3 x^2-3 x+2}$

$\lim _{x \rightarrow 0} \frac{x^2 2^x-x^2 \sin x-x^2}{3^x+\cos x-3^x \cos x-1}=$

The value of the limit of $\frac{x^3 - x^2 - 18}{x - 3}$ as $x$ tends to $3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module