[x] denotes the greatest integer less than or | vedclass

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

Question

(A) Given the limit $\lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=	heta$. For $x ightarrow 0^{-}$,we have $[x] = -1$ and $\{x\} =

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(A) Given the limit $\lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=	heta$. For $x ightarrow 0^{-}$,we have $[x] = -1$ and $\{x\} = x - [x] = x - (-1) = x+1$. Substituting these values into the expression: $	heta = \lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x-1)}{2-(x+1)} = \lim _{x ightarrow 0^{-}} \frac{\sin ^{-1}(x-1)}{1-x}$. As $x ightarrow 0^{-}$,the expression approaches $\frac{\sin ^{-1}(-1)}{1-0} = \frac{-\pi/2}{1} = -\pi/2$. Thus,$	heta = -\pi/2$. Now,we calculate $\sin 	heta + \cos 	heta = \sin(-\pi/2) + \cos(-\pi/2) = -1 + 0 = -1$.

$[x]$ denotes the greatest integer less than or equal to $x$. If $\{x\}=x-[x]$ and $\lim _{x \rightarrow 0^{-}} \frac{\sin ^{-1}(x+[x])}{2-\{x\}}=\theta$,then $\sin \theta+\cos \theta=$

Similar Questions

$\mathop {\lim }\limits_{x \to \frac{\pi^+}{2}} e^{[\cot x]}$ is equal to :-
(where $[.]$ is the greatest integer function)

If $\lim_{x \to 0} \frac{(4^x - 1)^3}{\tan(\frac{x}{4}) \log(1 + \frac{x^2}{3})} = 96(\log a)^b$, then $(a + b) = $

$\mathop {Limit}\limits_{x \to \frac{\pi }{2}} \,\frac{{\sin x}}{{{{\cos }^{ - 1}}\left[ {\frac{1}{4}\,(3\sin x\, - \,\sin 3x)} \right]}}\,$,where $[ \cdot ]$ denotes the greatest integer function,is

The value of $\mathop {\lim }\limits_{x \to 0} \left( \left[ \frac{100x}{\sin x} \right] + \left[ \frac{99 \sin x}{x} \right] \right)$,where $[.]$ denotes the greatest integer function,is

$\lim _{x \rightarrow 0} \frac{x^2 \sin ^2(3 x)+\sin ^4(6 x)}{(1-\cos 3 x)^2}=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module