If displaystyle limx → 0 (((a-n)nx - tan x ) sin nx/x2) = 0, where n is nonzero real number, then a is equal to

Question

If displaystyle limx → 0 (((a-n)nx - tan x ) sin nx/x2) = 0, where n is nonzero real number, then a is equal to (A) 0 (B) (n +1 /n) (C) n (D)  n + (1/n)

Tardigrade Admin · Accepted Answer

underset textx arrow 0 textlim((. (. a - n .) n textx - texttan x)sin textnx/( textx)2)=0 ⇒ underset textx arrow 0 textlim (( texta - textn) textn - ( texttan textx/ textx)) ( textsin textnx/ textx) = 0 ⇒ (( texta - textn) textn - 1) textn = 0 ⇒ ( texta - textn) textn - 1 = 0 (∵ textn ≠ 0) ∴ texta = textn + (1/ textn)

Q. If $x \to 0 lim \frac{(( a - n ) n x - tan x ) sin n x}{x ^{2}} = 0,$ where n is nonzero real number, then a is equal to

0

$\frac{n + 1}{n}$

n

$n + \frac{1}{n}$

$x \to 0 lim \frac{( ( a - n ) n x - tan x)sin nx}{( x ) ^{2}} = 0$
$\Rightarrow x \to 0 lim ((a - n) n - \frac{tan x}{x}) \frac{sin nx}{x} = 0$
$\Rightarrow ((a - n) n - 1) n = 0$
$\Rightarrow (a - n) n - 1 = 0$ $(∵ n \neq = 0)$
$∴ a = n + \frac{1}{n}$

Q. If x→0lim​x2((a−n)nx−tanx)sinnx​=0, where n is nonzero real number, then a is equal to

0

nn+1​

n

n+n1​

x→0lim​(x)2((a−n)nx−tan x)sin nx​=0 ⇒x→0lim​((a−n)n−xtan x​)xsin nx​=0 ⇒((a−n)n−1)n=0 ⇒(a−n)n−1=0 (∵n=0) ∴a=n+n1​

Q. If $x \to 0 lim \frac{(( a - n ) n x - tan x ) sin n x}{x ^{2}} = 0,$ where n is nonzero real number, then a is equal to

$\frac{n + 1}{n}$

$n + \frac{1}{n}$

$x \to 0 lim \frac{( ( a - n ) n x - tan x)sin nx}{( x ) ^{2}} = 0$
$\Rightarrow x \to 0 lim ((a - n) n - \frac{tan x}{x}) \frac{sin nx}{x} = 0$
$\Rightarrow ((a - n) n - 1) n = 0$
$\Rightarrow (a - n) n - 1 = 0$ $(∵ n \neq = 0)$
$∴ a = n + \frac{1}{n}$