_ h 0 (a + h) ^2 (a + h) - a^2 a h = - Vidyadip

MCQ

$\mathop {\lim }\limits_{h \to 0} \frac{{{{(a + h)}^2}\sin (a + h) - {a^2}\sin a}}{h} = $

A
$a\cos a + {a^2}\sin a$
B
$a\sin a + {a^2}\cos a$
✓
$2a\sin a + {a^2}\cos a$
D
$2a\cos a + {a^2}\sin a$

✓

Answer

Correct option: C.

$2a\sin a + {a^2}\cos a$

c
(c) $\frac{d}{{da}}\,[{a^2}\sin a] = 2a\sin a + {a^2}\cos a.$

Aliter : Apply $ L-$ Hospital’s rule,

$\mathop {\lim }\limits_{h \to 0} \,\frac{{{{(a + h)}^2}\sin (a + h) - {a^2}\sin a}}{h}$

$ = \mathop {\lim }\limits_{h \to 0} \,\,\frac{{2\,(a + h)\,\sin \,(a + h) + {{(a + h)}^2}\cos \,(a + h)}}{1}$

$ = 2a\,\,\sin a + {a^2}\cos \,\,a.$

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

There are $n$ white and $n$ black balls marked $1, 2, 3, ...., n$. The number of ways in which we can arrange these balls in $a$ row so that neighbouring balls are of different colours are

Let the relations $R_1$ and $R_2$ on the set
$X =\{1,2,3, \ldots, 20\}$ be given by
$R _1=\{( x , y ): 2 x -3 y =2\}$ and
$R_2=\{(x, y):-5 x+4 y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M + N$ equals

If the straight line $y = mx + c$ touches the circle ${x^2} + {y^2} - 2x - 4y + 3 = 0$ at the point $(2, 3)$, then $c =$

If for the complex numbers $z$ satisfying $|z-2-2 i| \leq 1$, the maximum value of $|3 i z+6|$ is attained at $\mathrm{a}+i \mathrm{~b}$, then $\mathrm{a}+\mathrm{b}$ is equal to .... .

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,{x^2},\,{\rm{when}}\,\,x \le 1\\x + 5,{\rm{when\,\, }}x > {\rm{1}}\end{array} \right.$, then

If a circle passes through the point $(1, 2)$ and cuts the circle ${x^2} + {y^2} = 4$ orthogonally, then the equation of the locus of its centre is

If $y = {\tan ^{ - 1}}\sqrt {{{1 + \cos x} \over {1 - \cos x}}} $, then ${{dy} \over {dx}}$ is equal to

Suppose four balls labelled $1,2,3,4$ are randomly placed in boxes $B_1, B_2, B_3, B_4$. The probability that exactly one box is empty is

A ray of light through $(2,1)$ is reflected at a point $P$ on the $y$ - axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of an ellipse with eccentrieity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{\sqrt{53}}$, then the equation of the other directrix can be :

The term independent of $x$ in the expansion of
$\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x+1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x>1$ is: