The value of the displaystyle limx→1 ( sin(ex-1-1)/ log x) is

Question

The value of the displaystyle limx→1 ( sin(ex-1-1)/ log x) is (A) 0 (B) e (C) (1/e) (D) 1

Tardigrade Admin · Accepted Answer

displaystyle limx→1 ( sin(ex-1-1)/ log x) = displaystyle limh→0 ( sin (eh-1)/ log (1+h)) = displaystyle limh→0 ( sin(eh-1)/ log (1+h)) = displaystyle limh→0 ( sin (eh-1)/(eh-1))×((eh-1)/ log (1+h)) =1× displaystyle limh→0 ((h+(h2/2!)+ ldots)/(h-(h2)2+ ldots∞)) =1×1=1

Q. The value of the $x \to 1 lim \frac{sin ( e ^{x - 1} - 1 )}{lo g x}$ is

$0$

$e$

$\frac{1}{e}$

$1$

Q. The value of the x→1lim​logxsin(ex−1−1)​ is

0

e

e1​

1

x→1lim​logxsin(ex−1−1)​ =h→0lim​log(1+h)sin(eh−1)​ =h→0lim​log(1+h)sin(eh−1)​ =h→0lim​(eh−1)sin(eh−1)​×log(1+h)(eh−1)​ =1×h→0lim​(h−2h2​+…∞)(h+2!h2​​+…)​ =1×1=1

Q. The value of the $x \to 1 lim \frac{sin ( e ^{x - 1} - 1 )}{lo g x}$ is

$0$

$e$

$\frac{1}{e}$

$1$