A calculus problem by Abdulrahman El Shafei

Calculus Level 2

$\large \lim_{ x\to \infty }\dfrac { \sqrt [ 3 ]{ 8{ x }^{ 3 }+1 } }{ | x | } = \, ?$

Notation: $| \cdot |$ denotes the absolute value function .

The answer is 2.0.

2 solutions

Chew-Seong Cheong
Dec 1, 2016

$\begin{aligned} L & = \lim_{x \to \infty} \frac {\sqrt[3]{8x^3+1}}{\color{#3D99F6}|x|} & \small \color{#3D99F6} \text{For }x > 0, \ |x| = x \\ & = \lim_{x \to \infty} \frac {\sqrt[3]{8x^3+1}}{\color{#3D99F6}x} & \small \color{#3D99F6} \text{Divide up and down by }x \\ & = \lim_{x \to \infty} \frac {\sqrt[3]{8+\frac 1{x^3}}}1 \\ & = \sqrt[3]{8} = \boxed{2} \end{aligned}$

Edwin Gray
Apr 9, 2019

Neglecting the 1 as x approaches infinity, limit approaches (2x)/x = 2.

A calculus problem by Abdulrahman El Shafei

The answer is 2.0.

2 solutions

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