What is the limit as x approaches 0+of (3xe^{2x})/(sin(6x)) ?

The limit as x approaches 0+of (3xe^{2x})/(sin(6x)) is 1/2

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limit as x approaches 0+of (3xe^{2x})/(sin(6x))

x \to 0 + lim (\frac{3 x e ^{2 x}}{sin ( 6 x )})

\frac{1}{2}

Decimal

0.5

Solution steps

x \to 0 + lim (\frac{3 x e ^{2 x}}{sin ( 6 x )})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 3 \cdot x \to 0 + lim (\frac{x e ^{2 x}}{sin ( 6 x )})

\frac{x e ^{2 x}}{sin ( 6 x )} = e^{2 x} \cdot \frac{x}{sin ( 6 x )}

= 3 \cdot x \to 0 + lim (e^{2 x} \frac{x}{sin ( 6 x )})

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

= 3 \cdot x \to 0 + lim (e^{2 x}) \cdot x \to 0 + lim (\frac{x}{sin ( 6 x )})

x \to 0 + lim (e^{2 x}) = 1

x \to 0 + lim (\frac{x}{sin ( 6 x )}) = \frac{1}{6}

= 3 \cdot 1 \cdot \frac{1}{6}

Simplify

3 \cdot 1 \cdot \frac{1}{6} : \frac{1}{2}

= \frac{1}{2}

What is the limit as x approaches 0+of (3xe^{2x})/(sin(6x)) ?
The limit as x approaches 0+of (3xe^{2x})/(sin(6x)) is 1/2