Download App

12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin (x + a) + \sin (a - x) - 2\sin a}}{{x\sin x}}} \right] = $
  • A
    $\sin a$
  • B
    $\cos a$
  • ✓
    $ - \sin a$
  • D
    $\frac{1}{2}\cos a$
✓

Answer

Correct option: C.
$ - \sin a$
c
(c) $\mathop {\lim }\limits_{x \to 0} \,\,2\,\sin \,a\,.\,\frac{{(\cos x - 1)}}{{x\sin x}}$

$ = - 2\,\sin a\,.\,\frac{{(1 - \cos x)}}{{{x^2}}}\,.\,\left( {\frac{x}{{\sin x}}} \right)$

$ = \mathop {\lim }\limits_{x \to 0} \,\, - 2\sin a\,.\,\frac{{2\,{{\sin }^2}(x/2)}}{{4\,{{\left( {\frac{x}{2}} \right)}^2}\,\left( {\frac{{\sin x}}{x}} \right)}} = - \sin 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

More "MCQ" from 12. limits→12. limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^{2} x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to
→
The most general value of $\theta $ which will satisfy both the equations $\sin \theta = - \frac{1}{2}$ and $\tan \theta = \frac{1}{{\sqrt 3 }}$ is
→
Lines $L_1: y-x=0$ and $L_2: 2 x+y=0$ intersect the line $L_3: y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.

$STATEMENT -1$ : The ratio $PR$: $RQ$ equals $2 \sqrt{2}: \sqrt{5}$. because

$STATEMENT -2$ : In any triangle, bisector of an angle divides the triangle into two similar triangles.

→
If $\mathop {\lim }\limits_{x \to a} \frac{{{x^9} + {a^9}}}{{x + a}} = 9$, then $a = $
→
If $|{z_1} + {z_2}| = |{z_1} - {z_2}|$, then the difference in the amplitudes of ${z_1}$ and ${z_2}$ is
→
If the ratio of the roots of $a{x^2} + 2bx + c = 0$ is same as the ratio of the roots of $p{x^2} + 2qx + r = 0$, then
→
If a circle and a square have the same perimeter, then
→
If $\text{A}\cap\text{B}=\text{B},$ then:
→
Let $f$ be a differentiable function and equation of normal to the graph of $y = f(x)$ at $x = 3$ is $3y = x + 18$ . If $L\, = \,\mathop {Lim}\limits_{x \to 1} \frac{{f\left( {3 + {{\left( {4{{\tan }^{ - 1}}x - \pi } \right)}^2}} \right) - f\left( {3 + {{\left( {f\left( 3 \right) - x - 6} \right)}^2}} \right)}}{{{{\sin }^2}\left( {x - 1} \right)}}$ then
→
Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statement is (are) $TRUE$?

$(A)$ The length of the line segment $O A_1$ is $4 \sqrt{3}$

$(B)$ The length of the line segment $A_1 B_1$ is 16

$(C)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(0,0)$

$(D)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(1,0)$

→