displaystyle lim x arrow 0 (ex-e-x-2 x/x- sin x) is equal to

Question

displaystyle lim x arrow 0 (ex-e-x-2 x/x- sin x) is equal to (A) 1 / 2 (B) 1 (C) 2 (D) 4

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 (ex-e-x-2 x/x- sin x) = displaystyle lim x arrow 0 (ex+e-x-2/1- cos x)(Using L'Hospital's Rule) = displaystyle lim x arrow 0 (ex-e-x/ sin x)(Using LâHospitalâs Rule) = displaystyle lim x arrow 0 (ex+e-x/ cos x) (Using LâHospitalâs Rule) =2

Q. $x \to 0 lim \frac{e ^{x} - e ^{- x} - 2 x}{x - sin x}$ is equal to

1 / 2

1

2

4

$x \to 0 lim \frac{e ^{x} - e ^{- x} - 2 x}{x - sin x}$
$= x \to 0 lim \frac{e ^{x} + e ^{- x} - 2}{1 - cos x}$ (Using L'Hospital's Rule)
$= x \to 0 lim \frac{e ^{x} - e ^{- x}}{sin x}$ (Using L’Hospital’s Rule)
$= x \to 0 lim \frac{e ^{x} + e ^{- x}}{cos x}$ (Using L’Hospital’s Rule)
$= 2$

Q. x→0lim​x−sinxex−e−x−2x​ is equal to

1 / 2

1

2

4

x→0lim​x−sinxex−e−x−2x​ =x→0lim​1−cosxex+e−x−2​(Using L'Hospital's Rule) =x→0lim​sinxex−e−x​(Using L’Hospital’s Rule) =x→0lim​cosxex+e−x​ (Using L’Hospital’s Rule) =2

Q. $x \to 0 lim \frac{e ^{x} - e ^{- x} - 2 x}{x - sin x}$ is equal to

$x \to 0 lim \frac{e ^{x} - e ^{- x} - 2 x}{x - sin x}$
$= x \to 0 lim \frac{e ^{x} + e ^{- x} - 2}{1 - cos x}$ (Using L'Hospital's Rule)
$= x \to 0 lim \frac{e ^{x} - e ^{- x}}{sin x}$ (Using L’Hospital’s Rule)
$= x \to 0 lim \frac{e ^{x} + e ^{- x}}{cos x}$ (Using L’Hospital’s Rule)
$= 2$