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

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

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limit as x approaches 0 of (tan(x/3))/x

x \to 0 lim (\frac{tan ( \frac{x}{3} )}{x})

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

x \to 0 lim (\frac{tan ( \frac{x}{3} )}{x})

Apply L'Hopital's Rule

: x \to 0 lim (\frac{sec ^{2} ( \frac{x}{3} ) \frac{1}{3}}{1})

= x \to 0 lim (\frac{sec ^{2} ( \frac{x}{3} ) \frac{1}{3}}{1})

Plug in the value

x = 0

= \frac{sec ^{2} ( \frac{0}{3} ) \frac{1}{3}}{1}

Simplify

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

= \frac{1}{3}

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\int_{- 1}^{1} \frac{1}{1 - x ^{2}} d x

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