Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

One to any power is one.

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

Step 1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} 4 x^{3} - x - 3}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

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

$\frac{0}{lim_{x \to 1} 4 x^{3} - lim_{x \to 1} x - lim_{x \to 1} 3}$

Step 1.3.2

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

$\frac{0}{4 lim_{x \to 1} x^{3} - lim_{x \to 1} x - lim_{x \to 1} 3}$

Step 1.3.3

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

$\frac{0}{4 {(lim_{x \to 1} x)}^{3} - lim_{x \to 1} x - lim_{x \to 1} 3}$

Step 1.3.4

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{0}{4 {(lim_{x \to 1} x)}^{3} - lim_{x \to 1} x - 1 \cdot 3}$

Step 1.3.5

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

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Step 1.3.5.1

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

$\frac{0}{4 \cdot 1^{3} - lim_{x \to 1} x - 1 \cdot 3}$

Step 1.3.5.2

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

$\frac{0}{4 \cdot 1^{3} - 1 - 1 \cdot 3}$

Step 1.3.6

Simplify the answer.

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Step 1.3.6.1

Simplify each term.

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Step 1.3.6.1.1

One to any power is one.

$\frac{0}{4 \cdot 1 - 1 - 1 \cdot 3}$

Step 1.3.6.1.2

Multiply $4$ by $1$ .

$\frac{0}{4 - 1 - 1 \cdot 3}$

Step 1.3.6.1.3

Multiply $- 1$ by $3$ .

$\frac{0}{4 - 1 - 3}$

Step 1.3.6.2

Subtract $1$ from $4$ .

$\frac{0}{3 - 3}$

Step 1.3.6.3

Subtract $3$ from $3$ .

$\frac{0}{0}$

Step 1.3.6.4

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

Undefined

$\frac{0}{0}$

Step 1.3.7

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of $x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1]$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.4

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

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

Step 3.5

Add $3 x^{2}$ and $0$ .

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

Step 3.6

By the Sum Rule, the derivative of $4 x^{3} - x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$ .

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

Step 3.7

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 3.7.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

Step 3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.7.3

Multiply $3$ by $4$ .

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]}$

Step 3.8

Evaluate $\frac{d}{d x} [- x]$ .

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Step 3.8.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} - \frac{d}{d x} [x] + \frac{d}{d x} [- 3]}$

Step 3.8.2

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

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} - 1 \cdot 1 + \frac{d}{d x} [- 3]}$

Step 3.8.3

Multiply $- 1$ by $1$ .

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} - 1 + \frac{d}{d x} [- 3]}$

Step 3.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} - 1 + 0}$

Step 3.10

Add $12 x^{2} - 1$ and $0$ .

$lim_{x \to 1} \frac{3 x^{2}}{12 x^{2} - 1}$

Step 4

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

$3 lim_{x \to 1} \frac{x^{2}}{12 x^{2} - 1}$

Step 5

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

$3 \frac{lim_{x \to 1} x^{2}}{lim_{x \to 1} 12 x^{2} - 1}$

Step 6

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

$3 \frac{{(lim_{x \to 1} x)}^{2}}{lim_{x \to 1} 12 x^{2} - 1}$

Step 7

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

$3 \frac{{(lim_{x \to 1} x)}^{2}}{lim_{x \to 1} 12 x^{2} - lim_{x \to 1} 1}$

Step 8

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{{(lim_{x \to 1} x)}^{2}}{12 lim_{x \to 1} x^{2} - lim_{x \to 1} 1}$

Step 9

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

$3 \frac{{(lim_{x \to 1} x)}^{2}}{12 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}$

Step 10

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

$3 \frac{{(lim_{x \to 1} x)}^{2}}{12 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 11

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

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Step 11.1

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

$3 \frac{1^{2}}{12 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 11.2

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

$3 \frac{1^{2}}{12 \cdot 1^{2} - 1 \cdot 1}$

Step 12

Simplify the answer.

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Step 12.1

One to any power is one.

$3 \frac{1}{12 \cdot 1^{2} - 1 \cdot 1}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

One to any power is one.

$3 \frac{1}{12 \cdot 1 - 1 \cdot 1}$

Step 12.2.2

Multiply $12$ by $1$ .

$3 \frac{1}{12 - 1 \cdot 1}$

Step 12.2.3

Multiply $- 1$ by $1$ .

$3 \frac{1}{12 - 1}$

Step 12.2.4

Subtract $1$ from $12$ .

$3 (\frac{1}{11})$

Step 12.3

Combine $3$ and $\frac{1}{11}$ .

$\frac{3}{11}$