What is the limit as x approaches 0 of (arcsin(5x))/(sin(3x)) ?

The limit as x approaches 0 of (arcsin(5x))/(sin(3x)) is 5/3

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limit as x approaches 0 of (arcsin(5x))/(sin(3x))

x \to 0 lim (\frac{arcsin ( 5 x )}{sin ( 3 x )})

\frac{5}{3}

Decimal

1.66666 \dots

Solution steps

x \to 0 lim (\frac{arcsin ( 5 x )}{sin ( 3 x )})

Apply L'Hopital's Rule

= x \to 0 lim (\frac{\frac{5}{1 - 25 x ^{2}}}{cos ( 3 x ) \cdot 3})

Plug in the value

x = 0

= \frac{\frac{5}{1 - 25 \cdot 0 ^{2}}}{cos ( 3 \cdot 0 ) \cdot 3}

Simplify

\frac{\frac{5}{1 - 25 \cdot 0 ^{2}}}{cos ( 3 \cdot 0 ) \cdot 3} : \frac{5}{3}

= \frac{5}{3}

What is the limit as x approaches 0 of (arcsin(5x))/(sin(3x)) ?
The limit as x approaches 0 of (arcsin(5x))/(sin(3x)) is 5/3