What is the limit as x approaches 0 of (arctan(x)-x)/(8x^3) ?

The limit as x approaches 0 of (arctan(x)-x)/(8x^3) is -1/24

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limit as x approaches 0 of (arctan(x)-x)/(8x^3)

x \to 0 lim (\frac{arctan ( x ) - x}{8 x ^{3}})

- \frac{1}{24}

Decimal Notation

- 0.04166 \dots

Solution steps

x \to 0 lim (\frac{arctan ( x ) - x}{8 x ^{3}})

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

= \frac{1}{8} \cdot x \to 0 lim (\frac{arctan ( x ) - x}{x ^{3}})

Apply L'Hopital's Rule

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

= \frac{1}{8} \cdot x \to 0 lim (\frac{\frac{1}{x ^{2} + 1} - 1}{3 x ^{2}})

Simplify

\frac{\frac{1}{x ^{2} + 1} - 1}{3 x ^{2}} : - \frac{1}{3 ( x ^{2} + 1 )}

= \frac{1}{8} \cdot x \to 0 lim (- \frac{1}{3 ( x ^{2} + 1 )})

Plug in the value

x = 0

= \frac{1}{8} (- \frac{1}{3 ( 0 ^{2} + 1 )})

Simplify

\frac{1}{8} (- \frac{1}{3 ( 0 ^{2} + 1 )}) : - \frac{1}{24}

= - \frac{1}{24}

\int 2 (sin (x) - cos (x) + x) d x

x \to - 2 lim (x^{2} - 3 x + 5)

\int_{2}^{x} 1 + t^{2} d t

y = (a x + b)^{6}

\int \frac{1}{x ^{2} + 16} d x

What is the limit as x approaches 0 of (arctan(x)-x)/(8x^3) ?
The limit as x approaches 0 of (arctan(x)-x)/(8x^3) is -1/24