What is the limit as x approaches 0 of (e^{3x}-1)/(sin(2x)) ?

The limit as x approaches 0 of (e^{3x}-1)/(sin(2x)) is 3/2

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limit as x approaches 0 of (e^{3x}-1)/(sin(2x))

x \to 0 lim (\frac{e ^{3 x} - 1}{sin ( 2 x )})

\frac{3}{2}

Decimal Notation

1.5

Solution steps

x \to 0 lim (\frac{e ^{3 x} - 1}{sin ( 2 x )})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{3 e ^{3 x}}{cos ( 2 x ) \cdot 2})

= x \to 0 lim (\frac{3 e ^{3 x}}{cos ( 2 x ) \cdot 2})

Plug in the value

x = 0

= \frac{3 e ^{3 \cdot 0}}{cos ( 2 \cdot 0 ) \cdot 2}

Simplify

\frac{3 e ^{3 \cdot 0}}{cos ( 2 \cdot 0 ) \cdot 2} : \frac{3}{2}

= \frac{3}{2}

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What is the limit as x approaches 0 of (e^{3x}-1)/(sin(2x)) ?
The limit as x approaches 0 of (e^{3x}-1)/(sin(2x)) is 3/2