What is the limit as x approaches 0 of (sin(2x))/(3x) ?

The limit as x approaches 0 of (sin(2x))/(3x) is 2/3

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limit as x approaches 0 of (sin(2x))/(3x)

x \to 0 lim (\frac{sin ( 2 x )}{3 x})

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

x \to 0 lim (\frac{sin ( 2 x )}{3 x})

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

= \frac{1}{3} \cdot x \to 0 lim (\frac{sin ( 2 x )}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( 2 x ) \cdot 2}{1})

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

Plug in the value

x = 0

= \frac{1}{3} \cdot \frac{cos ( 2 \cdot 0 ) \cdot 2}{1}

Simplify

\frac{1}{3} \cdot \frac{cos ( 2 \cdot 0 ) \cdot 2}{1} : \frac{2}{3}

= \frac{2}{3}

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\int x 4 x^{2} - 5 d x

f (x) = (3 x^{4} + 8)^{\frac{4}{3}}

What is the limit as x approaches 0 of (sin(2x))/(3x) ?
The limit as x approaches 0 of (sin(2x))/(3x) is 2/3