_ x 0 x x - x x^2 x = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \frac{{x\cos x - \sin x}}{{{x^2}\sin x}} = $

A
$\frac{1}{3}$
✓
$ - \frac{1}{3}$
C
$1$
D
None of these

✓

Answer

Correct option: B.

$ - \frac{1}{3}$

b
(b) $\mathop {\lim }\limits_{x \to 0} \,\,\frac{{x\cos x - \sin x}}{{{x^2}\sin x}}$

$ = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{ - \sin x}}{{2\sin x + x\cos x}}$

(By $L-$ Hospital’s rule)

$ = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{ - \cos x}}{{3\cos x - x\sin x}} = - \frac{1}{3}$,

(Again by $L-$ Hospital’s rule)

$ = - \frac{1}{3}$

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

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, parallel to the straight line $2 x-y=1$. The points of contacts of the tangents on the hyperbola are

$(A)$ $\left(\frac{9}{2 \sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ $(B)$ $\left(-\frac{9}{2 \sqrt{2}},-\frac{1}{\sqrt{2}}\right)$

$(C)$ $(3 \sqrt{3},-2 \sqrt{2})$ $(D)$ $(-3 \sqrt{3}, 2 \sqrt{2})$

Four distinct points $(2 \mathrm{k}, 3 \mathrm{k}),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+5\, \hat{\mathrm{j}}+\alpha\, \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+3 \,\hat{\mathrm{j}}+\beta\, \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=-\hat{\mathrm{i}}+2\, \hat{\mathrm{j}}-3 \,\hat{\mathrm{k}}$ be three vectors such that, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=5 \sqrt{3}$ and $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{b}}$. Then the greatest amongst the values of $|\vec{a}|^{2}$ is .... .

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to

The number of ways in which ten candidates ${A_1},\;{A_2},\;.......{A_{10}}$ can be ranked such that ${A_1}$ is always above ${A_{10}}$ is

The value of ${\sin ^{ - 1}}(\sin 10)$ is

Let $S$ be the set of all $(\alpha, \beta), \pi<\alpha, \beta<2 \pi$, for which the complex number $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely imaginary and $\frac{1+i \cos \beta}{1-2 i \cos \beta}$ is purely real. Let $Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$ . Then $\sum_{(\alpha, \beta) \in s }\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to.

A particle moves in a straight line in such a way that its velocity at any point is given by ${v^2} = 2 - 3x$, where $x$ is measured from a fixed point. The acceleration is

$\int_{}^{} {\sqrt {1 - \sin 2x} \;} dx = ........,\;\;x \in (0,\;\pi /4)$

If $|a|\, = 3,\,\,|b|\, = 4$ then a value of $\lambda$ for which $a + \lambda b$ is perpendicular to $a - \lambda b$ is