Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x^{3} - 1}{x^{3} - 1}$ .

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

Step 1.2

To write $- \frac{3}{x^{3} - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 1.3

Write each expression with a common denominator of $(x - 1) (x^{3} - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{x - 1}$ by $\frac{x^{3} - 1}{x^{3} - 1}$ .

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

Step 1.3.2

Multiply $\frac{3}{x^{3} - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.3.3

Reorder the factors of $(x^{3} - 1) (x - 1)$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

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.

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

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

Step 2.1.2.7.2

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

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

Step 2.1.2.8

Simplify the answer.

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Step 2.1.2.8.1

Simplify each term.

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Step 2.1.2.8.1.1

One to any power is one.

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

Step 2.1.2.8.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.2.8.1.3

Multiply $- 1$ by $1$ .

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

Step 2.1.2.8.1.4

Subtract $1$ from $1$ .

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

Step 2.1.2.8.1.5

Multiply $- 3$ by $0$ .

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

Step 2.1.2.8.2

Subtract $1$ from $1$ .

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

Step 2.1.2.8.3

Add $0$ and $0$ .

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

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 $1$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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

Step 2.1.3.6

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

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

Step 2.1.3.7

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

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Step 2.1.3.7.1

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

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

Step 2.1.3.7.2

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

$\frac{0}{(1 - 1 \cdot 1) \cdot (1^{3} - 1 \cdot 1)}$

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Multiply $- 1$ by $1$ .

$\frac{0}{(1 - 1) \cdot (1^{3} - 1 \cdot 1)}$

Step 2.1.3.8.2

Subtract $1$ from $1$ .

$\frac{0}{0 \cdot (1^{3} - 1 \cdot 1)}$

Step 2.1.3.8.3

Simplify each term.

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Step 2.1.3.8.3.1

One to any power is one.

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

Step 2.1.3.8.3.2

Multiply $- 1$ by $1$ .

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

Step 2.1.3.8.4

Subtract $1$ from $1$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.8.5

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.8.6

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

Undefined

$\frac{0}{0}$

Step 2.1.3.9

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Evaluate $\frac{d}{d x} [- 3 (x - 1)]$ .

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Step 2.3.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 (x - 1)$ with respect to $x$ is $- 3 \frac{d}{d x} [x - 1]$ .

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

Step 2.3.5.2

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

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

Step 2.3.5.3

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

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

Step 2.3.5.4

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

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

Step 2.3.5.5

Add $1$ and $0$ .

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

Step 2.3.5.6

Multiply $- 3$ by $1$ .

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

Step 2.3.6

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

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

Step 2.3.7

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 $f (x) = x - 1$ and $g (x) = x^{3} - 1$ .

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

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

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

Step 2.3.11

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

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

Step 2.3.12

Move $3$ to the left of $x - 1$ .

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

Step 2.3.13

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

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

Step 2.3.14

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

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

Step 2.3.15

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

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

Step 2.3.16

Add $1$ and $0$ .

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

Step 2.3.17

Multiply $x^{3} - 1$ by $1$ .

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

Step 2.3.18

Simplify.

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Step 2.3.18.1

Apply the distributive property.

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

Step 2.3.18.2

Apply the distributive property.

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

Step 2.3.18.3

Combine terms.

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Step 2.3.18.3.1

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

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Step 2.3.18.3.1.1

Move $x^{2}$ .

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

Step 2.3.18.3.1.2

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

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Step 2.3.18.3.1.2.1

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

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

Step 2.3.18.3.1.2.2

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

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

Step 2.3.18.3.1.3

Add $2$ and $1$ .

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

Step 2.3.18.3.2

Multiply $3$ by $- 1$ .

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

Step 2.3.18.3.3

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

One to any power is one.

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

Step 3.1.2.3.1.2

Multiply $3$ by $1$ .

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

Step 3.1.2.3.1.3

Multiply $- 1$ by $3$ .

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

Step 3.1.2.3.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to 1} 4 x^{3} - 3 x^{2} - 1}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

$\frac{0}{lim_{x \to 1} 4 x^{3} - lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 1}$

Step 3.1.3.2

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

$\frac{0}{4 lim_{x \to 1} x^{3} - lim_{x \to 1} 3 x^{2} - lim_{x \to 1} 1}$

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

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

$\frac{0}{4 {(lim_{x \to 1} x)}^{3} - 3 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 3.1.3.7

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

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Step 3.1.3.7.1

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

$\frac{0}{4 \cdot 1^{3} - 3 {(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 3.1.3.7.2

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

$\frac{0}{4 \cdot 1^{3} - 3 \cdot 1^{2} - 1 \cdot 1}$

Step 3.1.3.8

Simplify the answer.

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Step 3.1.3.8.1

Simplify each term.

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Step 3.1.3.8.1.1

One to any power is one.

$\frac{0}{4 \cdot 1 - 3 \cdot 1^{2} - 1 \cdot 1}$

Step 3.1.3.8.1.2

Multiply $4$ by $1$ .

$\frac{0}{4 - 3 \cdot 1^{2} - 1 \cdot 1}$

Step 3.1.3.8.1.3

One to any power is one.

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

Step 3.1.3.8.1.4

Multiply $- 3$ by $1$ .

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

Step 3.1.3.8.1.5

Multiply $- 1$ by $1$ .

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

Step 3.1.3.8.2

Subtract $3$ from $4$ .

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

Step 3.1.3.8.3

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 3.1.3.8.4

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

Undefined

$\frac{0}{0}$

Step 3.1.3.9

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

By the Sum Rule, the derivative of $3 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 3]$ .

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

Step 3.3.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 3.3.3.2

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

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

Step 3.3.3.3

Multiply $2$ by $3$ .

$lim_{x \to 1} \frac{6 x + \frac{d}{d x} [- 3]}{\frac{d}{d x} [4 x^{3} - 3 x^{2} - 1]}$

Step 3.3.4

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

$lim_{x \to 1} \frac{6 x + 0}{\frac{d}{d x} [4 x^{3} - 3 x^{2} - 1]}$

Step 3.3.5

Add $6 x$ and $0$ .

$lim_{x \to 1} \frac{6 x}{\frac{d}{d x} [4 x^{3} - 3 x^{2} - 1]}$

Step 3.3.6

By the Sum Rule, the derivative of $4 x^{3} - 3 x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{6 x}{\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.7

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

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Step 3.3.7.1

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

$lim_{x \to 1} \frac{6 x}{4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.7.2

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

$lim_{x \to 1} \frac{6 x}{4 (3 x^{2}) + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.7.3

Multiply $3$ by $4$ .

$lim_{x \to 1} \frac{6 x}{12 x^{2} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.8

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 3.3.8.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 1} \frac{6 x}{12 x^{2} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.8.2

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

$lim_{x \to 1} \frac{6 x}{12 x^{2} - 3 (2 x) + \frac{d}{d x} [- 1]}$

Step 3.3.8.3

Multiply $2$ by $- 3$ .

$lim_{x \to 1} \frac{6 x}{12 x^{2} - 6 x + \frac{d}{d x} [- 1]}$

Step 3.3.9

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

$lim_{x \to 1} \frac{6 x}{12 x^{2} - 6 x + 0}$

Step 3.3.10

Add $12 x^{2} - 6 x$ and $0$ .

$lim_{x \to 1} \frac{6 x}{12 x^{2} - 6 x}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to 1} \frac{x}{12 x^{2} - 6 x}$

Step 4.2

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

$6 \frac{lim_{x \to 1} x}{lim_{x \to 1} 12 x^{2} - 6 x}$

Step 4.3

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

$6 \frac{lim_{x \to 1} x}{lim_{x \to 1} 12 x^{2} - lim_{x \to 1} 6 x}$

Step 4.4

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

$6 \frac{lim_{x \to 1} x}{12 lim_{x \to 1} x^{2} - lim_{x \to 1} 6 x}$

Step 4.5

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

$6 \frac{lim_{x \to 1} x}{12 {(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 6 x}$

Step 4.6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 \frac{lim_{x \to 1} x}{12 {(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x}$

Step 5

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

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Step 5.1

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

$6 \frac{1}{12 {(lim_{x \to 1} x)}^{2} - 6 lim_{x \to 1} x}$

Step 5.2

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

$6 \frac{1}{12 \cdot 1^{2} - 6 lim_{x \to 1} x}$

Step 5.3

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

$6 \frac{1}{12 \cdot 1^{2} - 6 \cdot 1}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

One to any power is one.

$6 \frac{1}{12 \cdot 1 - 6 \cdot 1}$

Step 6.1.2

Multiply $12$ by $1$ .

$6 \frac{1}{12 - 6 \cdot 1}$

Step 6.1.3

Multiply $- 6$ by $1$ .

$6 \frac{1}{12 - 6}$

Step 6.1.4

Subtract $6$ from $12$ .

$6 (\frac{1}{6})$

Step 6.2

Cancel the common factor of $6$ .

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Step 6.2.1

Cancel the common factor.

$6 (\frac{1}{6})$

Step 6.2.2

Rewrite the expression.

$1$