What is the limit as x approaches 0 of (tan(2x))/(6x) ?

The limit as x approaches 0 of (tan(2x))/(6x) is 1/3

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limit as x approaches 0 of (tan(2x))/(6x)

x \to 0 lim (\frac{tan ( 2 x )}{6 x})

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

x \to 0 lim (\frac{tan ( 2 x )}{6 x})

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

= \frac{1}{6} \cdot x \to 0 lim (\frac{tan ( 2 x )}{x})

Apply L'Hopital's Rule

= \frac{1}{6} \cdot x \to 0 lim (\frac{sec ^{2} ( 2 x ) \cdot 2}{1})

Plug in the value

x = 0

= \frac{1}{6} \cdot \frac{sec ^{2} ( 2 \cdot 0 ) \cdot 2}{1}

Simplify

\frac{1}{6} \cdot \frac{sec ^{2} ( 2 \cdot 0 ) \cdot 2}{1} : \frac{1}{3}

= \frac{1}{3}

What is the limit as x approaches 0 of (tan(2x))/(6x) ?
The limit as x approaches 0 of (tan(2x))/(6x) is 1/3