_ x 0 ^ - 1 x - ^ - 1 x x^3 is equal to - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x}}{{{x^3}}}$ is equal to

A
$0$
B
$1$
C
$-1$
✓
$1/2$

✓

Answer

Correct option: D.

$1/2$

d
(d) $\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x}}{{{x^3}}}$, $\left( {\frac{0}{0}} \,\,form \,\, \right)$

Applying $ L-$ Hospital’s rule,

$\mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{\sqrt {1 - {x^2}} }} - \frac{1}{{1 + {x^2}}}}}{{3{x^2}}}$, $\left( {\frac{0}{0}} \,\,form \,\, \right)$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{{ - 1}}{2} \times \frac{{ - 2x}}{{{{(1 - {x^2})}^{3/2}}}} + \frac{{2x}}{{{{(1 + {x^2})}^2}}}}}{{6x}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{1}{6}\left[ {\frac{1}{{{{(1 - {x^2})}^{3/2}}}} + \frac{2}{{{{(1 + {x^2})}^2}}}} \right]\, = \,\frac{1}{2}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $x + by + c = 0$ is normal to parabola $y^2 = 12x$ , then complete set of all values of $c$ is

The sum of coefficients in ${(1 + x - 3{x^2})^{2134}}$ is

Let $f :[0,1] \rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f (0)=3$ and $f (1)=5$. If the line $y=2 x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$, then the least number of points $x \in(0,1)$, at which $f ^{\prime \prime}( x )=0$, is$......$

If $A = 1 + {r^z} + {r^{2z}} + {r^{3z}} + .......\infty $, then the value of $r$ will be

The area (in sq, units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1$ is :

$\mathop {\lim }\limits_{x \to a} \frac{{{{(x + 2)}^{5/3}} - {{(a + 2)}^{5/3}}}}{{x - a}} = $

The area of the parallelogram whose adjacent sides are $i - k$ and $2j + 3k$ is

Let $f$ and $g$ be twice differentiable even functions on $(-2,2)$ such that $f\left(\frac{1}{4}\right)=0, f\left(\frac{1}{2}\right)=0, f(1)=1$ and $g\left(\frac{3}{4}\right)=0, g(1)=2$ Then, the minimum number of solutions of $f(x) g^{\prime \prime}(x)+f^{\prime}(x) g^{\prime}(x)=0$ in $(-2,2)$ is equal to

Let $f ( x )=\left[2 x ^{2}+1\right]$ and $g ( x )=\left\{\begin{array}{ll}2 x -3, & x < 0 \\ 2 x +3, & x \geq 0\end{array}\right.$, where $[t]$ is the greatest integer $\leq t$ છે. Then, in the open interval $(-1,1)$, the number of points where fog is discontinuous is equal to

Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the arc AC , divides the arc $A C$ such that $\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1}{5}$, and $\overrightarrow{ OC }=\alpha \overrightarrow{ OA }+\beta \overrightarrow{ OB }$, then $\alpha=\sqrt{2}(\sqrt{3}-1) \beta$ is equal to