displaystyle lim x arrow 0 √(x- sin x/x+ sin 2 x) is equal to

Question

displaystyle lim x arrow 0 √(x- sin x/x+ sin 2 x) is equal to (A) 1 (B) 0 (C) ∞ (D) None of these

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 √(x- sin x/x+ sin 2 x) = displaystyle lim x arrow 0 √(1-( sin x/x)/1+( sin 2 x)x) = displaystyle lim x arrow 0 √(1-( sin x/x)/1+(( sin x)x) sin x) =√(1-1/1+1 × 0)=0

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

1

0

$\infty$

None of these

$x \to 0 lim \frac{x - sin x}{x + sin ^{2} x}$
$= x \to 0 lim \frac{1 - \frac{s i n x}{x}}{1 + \frac{s i n ^{2} x}{x}}$
$= x \to 0 lim \frac{1 - \frac{s i n x}{x}}{1 + ( \frac{s i n x}{x} ) sin x}$
$= \frac{1 - 1}{1 + 1 \times 0} = 0$

Q. x→0lim​x+sin2xx−sinx​​ is equal to

1

0

∞

None of these

x→0lim​x+sin2xx−sinx​​ =x→0lim​1+xsin2x​1−xsinx​​​ =x→0lim​1+(xsinx​)sinx1−xsinx​​​ =1+1×01−1​​=0

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

$\infty$

$x \to 0 lim \frac{x - sin x}{x + sin ^{2} x}$
$= x \to 0 lim \frac{1 - \frac{s i n x}{x}}{1 + \frac{s i n ^{2} x}{x}}$
$= x \to 0 lim \frac{1 - \frac{s i n x}{x}}{1 + ( \frac{s i n x}{x} ) sin x}$
$= \frac{1 - 1}{1 + 1 \times 0} = 0$