Evaluate: * lim _ x 0 tan x - sin x ^ 3 x . - Vidyadip

Question

Evaluate:

$\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\text{tan x - sin x}}{\sin^{3}\text{x}}$.

✓

Answer

$\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\text{tan x - sin x}}{\sin^{3}\text{x}}=\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\text{tan x}\cdot\text{(1 - cos x)}}{\sin^{3}\text{x}}$
=$\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\text{tan x}\text{(1 - cos x)}}{\text{x}^{3}\frac{\sin^{3}\text{x}}{\text{x}^{3}}}=\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}\cdot\frac{1 - \cos\text{x}}{\text{x}^{2}}\cdot\frac{1}{\frac{\sin^{3}\text{x}}{\text{x}^{3}}}$
$=\DeclareMathOperator*{\median}{\text{lim}} \median_{\text{x}\rightarrow0}\frac{\tan\text{x}}{\text{x}}\cdot\frac{\sin^{2}\text{x}}{\text{x}^{2}}\cdot\frac{1}{1+\text{cos x}}\cdot\frac{1}{\frac{\sin^{3}\text{x}}{\text{x}^{3}}}$
= $1.1\cdot\frac{1}{2}\cdot\frac{1}{1}=\frac{1}{2}$.

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

More "4 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&\text{if }0\leq\text{ x}<1\\\text{a},&\text{if }1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\text{if }\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous on $(0,\infty),$ then find the most suitable value of a and b.

If $\text{A}=\begin{bmatrix}4&2\\-1&-1 \end{bmatrix},$ prove that (A - 2I)(A - 3I) = 0

Find the inverse of the following matrices by using elementry row transformation:

$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 3 & -1 \\ 1 & -1 & 3 \end{bmatrix}$

Solve the differential equation $\frac{d y}{d x}+y \tan x=y^2$ $\sec x$.

Find which of the function:
$\text{f(x)}=\begin{cases}|\text{x}-\text{a}|\sin\frac{1}{\text{x}},&\text{if x}\neq0\\0,&\text{if x }=\text{a}\end{cases}$
at x = a

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum squares of its sides.

Solve the following differential equation:
$(1+\text{x}^2)\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{\tan^{-1}\text{x}}$

Evaluate the following integrals:
$\int\limits_{0}^{1}\text{x}\tan^{-1}\text{x}\text{ dx} $

Show that the following triads of vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-2\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}}$

Evaluate the following integrals:
$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$