The value of displaystyle lim x arrow 0 ((1+x)1 / x-e+(1/2) e x/x2) is

Question

The value of displaystyle lim x arrow 0 ((1+x)1 / x-e+(1/2) e x/x2) is (A) (-11 e/24) (B) (e/11) (C) (e/24) (D) (11 e/24)

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 ((1+x)1 / x-e+(1/2) e x/x2) displaystyle lim x arrow 0 (e(1-(x/2)+(11/24) x2+ ldots)-e+(1/2) e x/x2) =((11/24) e x2+ ldots ./x2)=(11/24) e

Q. The value of $x \to 0 lim \frac{( 1 + x ) ^{1/ x} - e + \frac{1}{2} e x}{x ^{2}}$ is

$\frac{- 11 e}{24}$

$\frac{e}{11}$

$\frac{e}{24}$

$\frac{11 e}{24}$

None of these

$x \to 0 lim \frac{( 1 + x ) ^{1/ x} - e + \frac{1}{2} e x}{x ^{2}}$
$x \to 0 lim \frac{e ( 1 - \frac{x}{2} + \frac{11}{24} x ^{2} + \dots ) - e + \frac{1}{2} e x}{x ^{2}}$
$= \frac{\frac{11}{24} e x ^{2} + \dots .}{x ^{2}} = \frac{11}{24} e$

Q. The value of x→0lim​x2(1+x)1/x−e+21​ex​ is

24−11e​

11e​

24e​

2411e​