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

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

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

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

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

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

= \frac{1}{3}

f (x) = - 3 x^{2} - 5 x + 1

\int cos^{4} (t) d t

y d y = 4 x (y^{2} + 1)^{\frac{1}{2}} d x

\int \frac{7 x ^{3} + 4 x}{x - 1} d x

y^{^{''}} + 25 y = - 40 sin (5 t)

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