Evaluate the following limit: _ x 1+ ^2 x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{1+\cos\text{x}}{\tan^2\text{ x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{1+\cos\text{x}}{\tan^2\text{ x}}$
$\Rightarrow\text{x}\rightarrow{\pi},\text{x}-{\pi}\rightarrow0,$ let $\text{y}=\text{x}-\pi$
$\Rightarrow\lim\limits_{\text{x}-{\pi}\rightarrow{{0}}}\frac{1+\cos\text{x}}{\tan^2\text{x}}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1+\cos(\pi+\text{y})}{\tan^2(\pi+\text{y})}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\cos\text{y}}{\tan^2\text{y}}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{2\sin^2\frac{\text{y}}{2}}{\tan^2\text{y}}$
$=2\Bigg(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\frac{\text{y}}{2}}{\frac{\text{y}}{2}}\Bigg)^2\times\frac{\text{y}^2}{4}\times\frac{1}{\Big(\lim\limits_{\text{y}\rightarrow{0}}\frac{\tan\text{y}}{\text{y}}\Big)^2\times\text{y}^2}$
$=2\times1\times\frac{\text{y}^2}{4}\times\frac{1}{1\times\text{y}^2}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1,\lim\limits_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1\Big]$
$=\frac{1}{2}$

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 "Solve the Following Question.(4 Marks)" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove the following identities:
$\frac{\cos\text{x}}{1-\sin\text{x}}=\frac{1+\cos\text{x}+\sin\text{x}}{1+\cos\text{x}-\sin\text{x}}$

Prove by the method of induction, for all n ∈ N.

$1^3+3^3+5^3+\ldots .$. to $n$ terms $=n^2\left(2 n^2-1\right)$

A number is drawn at random from the numbers 1 to 50. Find the probability that it is divisible by 2 or 3 or 10.

How many three letter words can be made using the letters of the word 'ORIENTAL'?

How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:

The letter G always occupies the first place?
The letters P and I respectively occupy first and last place?
The vowels are always together?
The vowels always occupy even places?

Sketch the graphs of the following functions:
$\text{f(x)}=\cot2\text{x}$

Solve the following system of equations in R.
$2\text{x}+6\geq0,4\text{x}-7<0$

Find the mean deviation from the median for the following data:

$x_i$	$15$	$21$	$27$	$30$
$f_i$	$3$	$5$	$6$	$7$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}}}{\text{x}\sqrt{\text{a}^2+\text{ax}}}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{ax}+\text{bx}}{\text{ax}+\sin\text{bx}}$