What is the limit as x approaches 0 of ((8+x)^{1/3}-2)/x ?

The limit as x approaches 0 of ((8+x)^{1/3}-2)/x is 1/12

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limit as x approaches 0 of ((8+x)^{1/3}-2)/x

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

\frac{1}{12}

Decimal Notation

0.08333 \dots

Solution steps

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

Apply L'Hopital's Rule

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

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

Plug in the value

x = 0

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

Simplify

\frac{\frac{1}{3 ( 0 + 8 ) ^{\frac{2}{3}}}}{1} : \frac{1}{12}

= \frac{1}{12}

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

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

x \to 0 lim (e^{\frac{1}{x}} - 1)

(y) \to (9) lim (3 x^{2} + 6 x y)

x \to 0 lim ((sec (x))^{4 x})

What is the limit as x approaches 0 of ((8+x)^{1/3}-2)/x ?
The limit as x approaches 0 of ((8+x)^{1/3}-2)/x is 1/12