Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

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

Step 2

Move the limit under the radical sign.

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

Step 3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

Move the limit under the radical sign.

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

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

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

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

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

Step 8.1.2

Any root of $1$ is $1$ .

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

Step 8.1.3

Any root of $1$ is $1$ .

$\frac{1 - 2 \cdot 1 + 1}{{(x^{2} - 1)}^{2}}$

Step 8.1.4

Multiply $- 2$ by $1$ .

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

Step 8.1.5

Subtract $2$ from $1$ .

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

Step 8.1.6

Add $- 1$ and $1$ .

$\frac{0}{{(x^{2} - 1)}^{2}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{0}{{(x^{2} - 1^{2})}^{2}}$

Step 8.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{0}{{((x + 1) (x - 1))}^{2}}$

Step 8.2.3

Apply the product rule to $(x + 1) (x - 1)$ .

$\frac{0}{{(x + 1)}^{2} {(x - 1)}^{2}}$

Step 8.3

Divide $0$ by ${(x + 1)}^{2} {(x - 1)}^{2}$ .

$0$