Calculus Examples

$lim_{x \to 0} \frac{1}{x^{\frac{1}{3}}} - \frac{1}{{(x - 1)}^{\frac{1}{3}}}$

Step 1

Convert fractional exponents to radicals.

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Step 1.1

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

$lim_{x \to 0} \frac{1}{\sqrt[3]{x}} - \frac{1}{{(x - 1)}^{\frac{1}{3}}}$

Step 1.2

Rewrite ${(x - 1)}^{\frac{1}{3}}$ as $\sqrt[3]{x - 1}$ .

$lim_{x \to 0} \frac{1}{\sqrt[3]{x}} - \frac{1}{\sqrt[3]{x - 1}}$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} \frac{1}{\sqrt[3]{x}} - \frac{1}{\sqrt[3]{x - 1}}$

Step 3

As the $x$ values approach $0$ from the left, the function values decrease without bound.

$- \infty$

Step 4

Consider the right sided limit.

$lim_{x \to 0^{+}} \frac{1}{\sqrt[3]{x}} - \frac{1}{\sqrt[3]{x - 1}}$

Step 5

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\infty$

Step 6

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$