Visit the website manishverma.site for latest posts, courses, admission & more.

### $\mathop {\lim }\limits_{x \to 0} {e^{\frac{{\tan x - \sin x}}{{{{\sin }^3}x}}}} = ?$

The given limit = ${e^{\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - \sin x}}{{{{\sin }^3}x}}}}$

$= {e^{\mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{\cos x}} - 1}}{{{{\sin }^2}x}}}}$

$= {e^{\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos x}}{{\cos x.{{\sin }^2}x}}}}$

$= {e^{\mathop {\lim }\limits_{x \to 0} \frac{{1 - \cos x}}{{\cos x.(1 - {{\cos }^2}x)}}}}$

$= {e^{\mathop {\lim }\limits_{x \to 0} \frac{1}{{\cos x.(1 + \cos x)}}}}$

$= {e^{1/2}} = \sqrt e$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$