Find the limit of ((x^3+1)^(1/3))/((x+1)^(1/3)) as x approaches -1

 Exercise

$\lim_{x\to-1}\left(\frac{\left(x^3+1\right)^{\frac{1}{3}}}{\left(x+1\right)^{\frac{1}{3}}}\right)$

 

Factor the sum or difference of cubes using the formula: $a^3\pm b^3 = (a\pm b)(a^2\mp ab+b^2)$

$\lim_{x\to-1}\left(\frac{\sqrt[3]{\left(x+1\right)\left(x^{2}-x+1\right)}}{\sqrt[3]{x+1}}\right)$

 

The power of a product is equal to the product of it's factors raised to the same power

$\lim_{x\to-1}\left(\frac{\sqrt[3]{x+1}\sqrt[3]{x^{2}-x+1}}{\sqrt[3]{x+1}}\right)$

 

Simplify the fraction $\frac{\sqrt{x+1}$

$\lim_{x\to-1}\left(\sqrt[3]{x^{2}-x+1}\right)$

 

Evaluate the limit $\lim_{x\to-1}\left(\sqrt[3]{x^{2}-x+1}\right)$ by replacing all occurrences of $x$ by $-1$

$\sqrt[3]{3}$

$\sqrt[3]{3}$

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