MCQ
$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{e^x} - {e^{\sin x}}}}{{x - \sin x}}} \right]$ is equal to
- A$-1$
- B$0$
- ✓$1$
- DNone of these
✓
Answer
Correct option: C.
$1$
c
(c) $\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{e^x} - {e^{\sin x}}}}{{x - \sin x}}} \right]$, $\left( {\frac{0}{0} \, \, \,{\rm{form}}} \right)$
(c) $\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{{e^x} - {e^{\sin x}}}}{{x - \sin x}}} \right]$, $\left( {\frac{0}{0} \, \, \,{\rm{form}}} \right)$
Using $ L-$ Hospital’s rule three times, then
$\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - {e^{\sin x}}.\cos x}}{{1 - \cos x}} = \mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - {e^{\sin x}}{{\cos }^2}x + \sin x.{e^{\sin x}}}}{{\sin x}}$
$ = \mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - {e^{\sin x}}.{{\cos }^3}x + {e^{\sin x}}2\cos x\sin x + {e^{\sin x}}.\cos x\sin x + {e^{\sin x}}.\cos x}}{{\cos x}}$
$ = 1$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Match the statements/expressions given in Column $I$ with the values given in Column $II$.
→| Column $I$ | Column $II$ |
|
$(A)$ Root$(s)$ of the equation $2 \sin ^2 \theta+\sin ^2 2 \theta=2$ |
$(p)$ $\frac{\pi}{6}$ |
|
$(B)$ Points of discontinuity of the function $f(x)=\left[\frac{6 x}{\pi}\right] \cos \left[\frac{3 x}{\pi}\right],$ where $[y]$ denotes the largest integer less than or equal to $y$ |
$(q)$ $\frac{\pi}{4}$ |
|
$(C)$ Volume of the parallelopiped with its edges represented by the vectors $\hat{i}+\hat{j}, \quad \hat{i}+2 \hat{j} \text { and } \hat{i}+\hat{j}+\pi \hat{k}$ |
$(r)$ $\frac{\pi}{3}$ |
|
$(D)$ Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors satisfying $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$ |
$(s)$ $\frac{\pi}{2}$ |
| $(t)$ $\pi$ |
The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :
→If graph of $y = ax^2 -bx + c$ is following, then sign of $a$, $b$, $c$ are
→The area of the region $S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\} \text { is }...... \,.$
→If $f(x) = \frac{{\sin \frac{{\pi x}}{4}}}{{x + 1}} ,$ then $\mathop {\lim }\limits_{h \to 0} \frac{{f(1 + h) - f(1)}}{{{h^2} + 2h}}$ is -
→If $a,\;b,\;c$ are in $A.P.$, then $\frac{{{{(a - c)}^2}}}{{({b^2} - ac)}} = $
→If $A, B, C$ are angles of a triangle, then $\sin 2A + \sin 2B - \sin 2C$ is equal to
→The eccentricity of the hyperbola $4{x^2} - 9{y^2} = 16$, is
→$\int_0^\pi {\frac{{xdx}}{{1 + \sin x}}} $ is equal to
→The number of all possible triplets $(a_1 , a_2 , a_3)$ such that $a_1+ a_2 \,cos \, 2x + a_3 \, sin^2 x = 0$ for all $x$ is
→