Evaluate the following limit: _ x 0 x^3 1- - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^3\cot\text{x}}{1-\cos\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^3\cot\text{x}}{1-\cos\text{x}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}^3}{\tan\text{x}(1-\cos\text{x})}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{\text{x}^3}{\tan\text{x}.2\sin^2\frac{\text{x}}{2}}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{1}{\frac{\tan\text{x}}{\text{x}}\times\frac{2\sin^2\frac{\text{x}}{2}}{\text{x}^2}}$
$=\frac{1}{\bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\tan\text{x}}{\text{x}}\bigg)\times2\bigg(\lim\limits_{\text{x} \rightarrow0}\frac{\sin\frac{\text{x}}{2}}{\frac{\text{x}}{2}}\bigg)^2\times\frac14}$
$=\frac{1}{2\times2\times1\times\frac14}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1,\lim\limits_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1\Big]$
$=2$

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 "3 Marks Question" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Show that the points (3, -2), (1, 0), (-1, -2) and (1, -4) are concyclic.

Find the mean and variance for each of the data

x_i	6	10	14	18	24	28	30
f_i	2	4	7	12	8	4	3

If $\text{y}=\Big(\frac{2-3\cos\text{x}}{\sin\text{x}}\Big),$ find $\frac{\text{dy}}{\text{dx}}$ at $\text{x}=\frac{\pi}{4}$

A cube of side 5 has one vertex at the point(1, 0, -1) and the three edge from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.

Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}2^{\text{n}-1}\sin\Big(\frac{\text{a}}{2^\text{n}}\Big)$

Coefficient of variation of the two distributions are 60 and 70 and their standard deviations are 21 and 16 respectively. What are their arithmetic means?

Find the equation of the straight line which meets the foot of perpendicular drawn from the origin to the lines $3 x-4 y=25$ and $3 x+5 y=17$.

Differentiate the following function with respect to x:

$\frac{\text{x}^2\cos\frac{\pi}{4}}{\sin\text{x}}$

Solve the following system of equations in R.
$\frac{|\text{x}-2|}{\text{x}-2}>0$

Find the equation of the circle passing through the points:
(5, 7), (8, 1) and (1, 3)