Calculus Examples

$lim_{x \to 1} \frac{x^{2} - x^{- 2}}{x - x^{- 1}}$

Step 1

Simplify terms.

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Step 1.1

Simplify the limit argument.

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Step 1.1.1

Convert negative exponents to fractions.

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Step 1.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 1} \frac{x^{2} - \frac{1}{x^{2}}}{x - x^{- 1}}$

Step 1.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 1} \frac{x^{2} - \frac{1}{x^{2}}}{x - \frac{1}{x}}$

Step 1.1.2

Combine terms.

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Step 1.1.2.1

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$lim_{x \to 1} \frac{\frac{x^{2} x^{2}}{x^{2}} - \frac{1}{x^{2}}}{x - \frac{1}{x}}$

Step 1.1.2.2

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{x^{2} x^{2} - 1}{x^{2}}}{x - \frac{1}{x}}$

Step 1.1.2.3

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 1} \frac{\frac{x^{2} x^{2} - 1}{x^{2}}}{\frac{x \cdot x}{x} - \frac{1}{x}}$

Step 1.1.2.4

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{x^{2} x^{2} - 1}{x^{2}}}{\frac{x \cdot x - 1}{x}}$

Step 1.2

Simplify the limit argument.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{x^{2} x^{2} - 1}{x^{2}} \cdot \frac{x}{x \cdot x - 1}$

Step 1.2.2

Combine factors.

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Step 1.2.2.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.2.2.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 1} \frac{x^{2 + 2} - 1}{x^{2}} \cdot \frac{x}{x \cdot x - 1}$

Step 1.2.2.1.2

Add $2$ and $2$ .

$lim_{x \to 1} \frac{x^{4} - 1}{x^{2}} \cdot \frac{x}{x \cdot x - 1}$

Step 1.2.2.2

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{x^{4} - 1}{x^{2}} \cdot \frac{x}{x^{1} x - 1}$

Step 1.2.2.3

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{x^{4} - 1}{x^{2}} \cdot \frac{x}{x^{1} x^{1} - 1}$

Step 1.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 1} \frac{x^{4} - 1}{x^{2}} \cdot \frac{x}{x^{1 + 1} - 1}$

Step 1.2.2.5

Add $1$ and $1$ .

$lim_{x \to 1} \frac{x^{4} - 1}{x^{2}} \cdot \frac{x}{x^{2} - 1}$

Step 1.2.2.6

Multiply $\frac{x^{4} - 1}{x^{2}}$ by $\frac{x}{x^{2} - 1}$ .

$lim_{x \to 1} \frac{(x^{4} - 1) x}{x^{2} (x^{2} - 1)}$

Step 1.2.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.2.3.1

Factor $x$ out of $(x^{4} - 1) x$ .

$lim_{x \to 1} \frac{x (x^{4} - 1)}{x^{2} (x^{2} - 1)}$

Step 1.2.3.2

Cancel the common factors.

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Step 1.2.3.2.1

Factor $x$ out of $x^{2} (x^{2} - 1)$ .

$lim_{x \to 1} \frac{x (x^{4} - 1)}{x (x (x^{2} - 1))}$

Step 1.2.3.2.2

Cancel the common factor.

$lim_{x \to 1} \frac{x (x^{4} - 1)}{x (x (x^{2} - 1))}$

Step 1.2.3.2.3

Rewrite the expression.

$lim_{x \to 1} \frac{x^{4} - 1}{x (x^{2} - 1)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 1} x^{4} - 1}{lim_{x \to 1} x (x^{2} - 1)}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$\frac{lim_{x \to 1} x^{4} - lim_{x \to 1} 1}{lim_{x \to 1} x (x^{2} - 1)}$

Step 2.1.2.1.2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{4} - lim_{x \to 1} 1}{lim_{x \to 1} x (x^{2} - 1)}$

Step 2.1.2.1.3

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

$\frac{{(lim_{x \to 1} x)}^{4} - 1 \cdot 1}{lim_{x \to 1} x (x^{2} - 1)}$

Step 2.1.2.2

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

$\frac{1^{4} - 1 \cdot 1}{lim_{x \to 1} x (x^{2} - 1)}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

One to any power is one.

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

Step 2.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.2.3.2

Subtract $1$ from $1$ .

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$\frac{0}{lim_{x \to 1} x \cdot lim_{x \to 1} x^{2} - 1}$

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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Step 2.1.3.5.1

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

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

Step 2.1.3.5.2

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

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

Step 2.1.3.6

Simplify the answer.

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Step 2.1.3.6.1

Multiply $1^{2} - 1 \cdot 1$ by $1$ .

$\frac{0}{1^{2} - 1 \cdot 1}$

Step 2.1.3.6.2

Simplify each term.

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Step 2.1.3.6.2.1

One to any power is one.

$\frac{0}{1 - 1 \cdot 1}$

Step 2.1.3.6.2.2

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 2.1.3.6.3

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 2.1.3.6.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.3.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{x^{4} - 1}{x (x^{2} - 1)} = lim_{x \to 1} \frac{\frac{d}{d x} [x^{4} - 1]}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{4} - 1]}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3.2

By the Sum Rule, the derivative of $x^{4} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$lim_{x \to 1} \frac{4 x^{3} + \frac{d}{d x} [- 1]}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{4 x^{3} + 0}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3.5

Add $4 x^{3}$ and $0$ .

$lim_{x \to 1} \frac{4 x^{3}}{\frac{d}{d x} [x (x^{2} - 1)]}$

Step 2.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} - 1$ .

$lim_{x \to 1} \frac{4 x^{3}}{x \frac{d}{d x} [x^{2} - 1] + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.7

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{4 x^{3}}{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 1} \frac{4 x^{3}}{x (2 x + \frac{d}{d x} [- 1]) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.9

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{4 x^{3}}{x (2 x + 0) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.10

Add $2 x$ and $0$ .

$lim_{x \to 1} \frac{4 x^{3}}{x (2 x) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.11

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{4 x^{3}}{2 (x^{1} x) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.12

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{4 x^{3}}{2 (x^{1} x^{1}) + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 1} \frac{4 x^{3}}{2 x^{1 + 1} + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.14

Add $1$ and $1$ .

$lim_{x \to 1} \frac{4 x^{3}}{2 x^{2} + (x^{2} - 1) \frac{d}{d x} [x]}$

Step 2.3.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{4 x^{3}}{2 x^{2} + (x^{2} - 1) \cdot 1}$

Step 2.3.16

Multiply $x^{2} - 1$ by $1$ .

$lim_{x \to 1} \frac{4 x^{3}}{2 x^{2} + x^{2} - 1}$

Step 2.3.17

Add $2 x^{2}$ and $x^{2}$ .

$lim_{x \to 1} \frac{4 x^{3}}{3 x^{2} - 1}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to 1} \frac{x^{3}}{3 x^{2} - 1}$

Step 3.2

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

$4 \frac{lim_{x \to 1} x^{3}}{lim_{x \to 1} 3 x^{2} - 1}$

Step 3.3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$4 \frac{{(lim_{x \to 1} x)}^{3}}{lim_{x \to 1} 3 x^{2} - 1}$

Step 3.4

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

$4 \frac{{(lim_{x \to 1} x)}^{3}}{lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 1}$

Step 3.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$4 \frac{{(lim_{x \to 1} x)}^{3}}{3 lim_{x \to 1} x^{2} - lim_{x \to 1} 1}$

Step 3.6

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

$4 \frac{{(lim_{x \to 1} x)}^{3}}{3 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}$

Step 3.7

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

$4 \frac{{(lim_{x \to 1} x)}^{3}}{3 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 4

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

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Step 4.1

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

$4 \frac{1^{3}}{3 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 4.2

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

$4 \frac{1^{3}}{3 \cdot 1^{2} - 1 \cdot 1}$

Step 5

Simplify the answer.

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Step 5.1

One to any power is one.

$4 \frac{1}{3 \cdot 1^{2} - 1 \cdot 1}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

One to any power is one.

$4 \frac{1}{3 \cdot 1 - 1 \cdot 1}$

Step 5.2.2

Multiply $3$ by $1$ .

$4 \frac{1}{3 - 1 \cdot 1}$

Step 5.2.3

Multiply $- 1$ by $1$ .

$4 \frac{1}{3 - 1}$

Step 5.2.4

Subtract $1$ from $3$ .

$4 (\frac{1}{2})$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Factor $2$ out of $4$ .

$2 (2) \frac{1}{2}$

Step 5.3.2

Cancel the common factor.

$2 \cdot 2 \frac{1}{2}$

Step 5.3.3

Rewrite the expression.

$2$