What is the limit as x approaches 0 of (sin(2x))/(arcsin(x)) ?

The limit as x approaches 0 of (sin(2x))/(arcsin(x)) is 2

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limit as x approaches 0 of (sin(2x))/(arcsin(x))

x \to 0 lim (\frac{sin ( 2 x )}{arcsin ( x )})

2

Solution steps

x \to 0 lim (\frac{sin ( 2 x )}{arcsin ( x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{cos ( 2 x ) \cdot 2}{\frac{1}{1 - x ^{2}}})

Plug in the value

x = 0

= \frac{cos ( 2 \cdot 0 ) \cdot 2}{\frac{1}{1 - 0 ^{2}}}

Simplify

\frac{cos ( 2 \cdot 0 ) \cdot 2}{\frac{1}{1 - 0 ^{2}}} : 2

= 2

What is the limit as x approaches 0 of (sin(2x))/(arcsin(x)) ?
The limit as x approaches 0 of (sin(2x))/(arcsin(x)) is 2