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

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

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limit as x approaches 0 of 1/x sin(x/3)

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

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

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

Simplify

\frac{1}{x} sin (\frac{x}{3}) : \frac{sin ( \frac{x}{3} )}{x}

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

Apply L'Hopital's Rule

: x \to 0 lim (\frac{cos ( \frac{x}{3} ) \frac{1}{3}}{1})

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

Plug in the value

x = 0

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

Simplify

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

= \frac{1}{3}

\frac{d y}{d x} + 7 y = 5

- 2 x^{2} (x + 4)

\frac{\partial}{\partial x} (4 (x y)^{2})

\frac{d}{d x} ((\frac{1}{x ^{2}} - \frac{9}{x ^{4}}) (x + 7 x^{3}))

\int_{- 1}^{1} x^{2} - 2 x + 3 d x

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