Calculus Examples

$lim_{x \to 1} \frac{{(x - 1)}^{5}}{x^{5} - 1}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 1} {(x - 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Move the exponent $5$ from ${(x - 1)}^{5}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x - 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.1.2

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

$\frac{{(lim_{x \to 1} x - lim_{x \to 1} 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.1.3

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

$\frac{{(lim_{x \to 1} x - 1 \cdot 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.2

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

$\frac{{(1 - 1 \cdot 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $1$ .

$\frac{{(1 - 1)}^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0^{5}}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.2.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{lim_{x \to 1} x^{5} - 1}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to 1} x^{5} - lim_{x \to 1} 1}$

Step 1.1.3.1.2

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x)}^{5} - lim_{x \to 1} 1}$

Step 1.1.3.1.3

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

$\frac{0}{{(lim_{x \to 1} x)}^{5} - 1 \cdot 1}$

Step 1.1.3.2

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

$\frac{0}{1^{5} - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

One to any power is one.

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

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 1} \frac{{(x - 1)}^{5}}{x^{5} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [{(x - 1)}^{5}]}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [{(x - 1)}^{5}]}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x - 1$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{\frac{d}{d u} [u^{5}] \frac{d}{d x} [x - 1]}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$lim_{x \to 1} \frac{5 u^{4} \frac{d}{d x} [x - 1]}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4} \frac{d}{d x} [x - 1]}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.3

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

$lim_{x \to 1} \frac{5 {(x - 1)}^{4} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.4

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

$lim_{x \to 1} \frac{5 {(x - 1)}^{4} (1 + \frac{d}{d x} [- 1])}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.5

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

$lim_{x \to 1} \frac{5 {(x - 1)}^{4} (1 + 0)}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.6

Add $1$ and $0$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4} \cdot 1}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.7

Multiply $5$ by $1$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{\frac{d}{d x} [x^{5} - 1]}$

Step 1.3.8

By the Sum Rule, the derivative of $x^{5} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 1]}$

Step 1.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{5 x^{4} + \frac{d}{d x} [- 1]}$

Step 1.3.10

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

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{5 x^{4} + 0}$

Step 1.3.11

Add $5 x^{4}$ and $0$ .

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{5 x^{4}}$

Step 1.4

Cancel the common factor of $5$ .

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Step 1.4.1

Cancel the common factor.

$lim_{x \to 1} \frac{5 {(x - 1)}^{4}}{5 x^{4}}$

Step 1.4.2

Rewrite the expression.

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

Move the exponent $4$ from ${(x - 1)}^{4}$ outside the limit using the Limits Power Rule.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$\frac{{(1 - 1 \cdot 1)}^{4}}{1^{4}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $1$ .

$\frac{{(1 - 1)}^{4}}{1^{4}}$

Step 4.1.2

Subtract $1$ from $1$ .

$\frac{0^{4}}{1^{4}}$

Step 4.1.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{1^{4}}$

Step 4.2

One to any power is one.

$\frac{0}{1}$

Step 4.3

Divide $0$ by $1$ .

$0$