Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

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.

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

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 $1$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

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 $x$ .

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

Step 2.1.2.7.2

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

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

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.

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

Step 2.1.2.8.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.2.8.1.3

Subtract $1$ from $1$ .

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

Step 2.1.2.8.1.4

Multiply $- 3$ by $0$ .

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

Step 2.1.2.8.2

Subtract $1$ from $1$ .

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

Step 2.1.2.8.3

Add $0$ and $0$ .

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

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 $1$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

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

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

Step 2.1.3.6

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

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

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 $x$ .

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

Step 2.1.3.7.2

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

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

Step 2.1.3.8

Simplify the answer.

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Step 2.1.3.8.1

Subtract $1$ from $1$ .

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

Step 2.1.3.8.2

Simplify each term.

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Step 2.1.3.8.2.1

One to any power is one.

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

Step 2.1.3.8.2.2

Multiply $- 1$ by $1$ .

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

Step 2.1.3.8.3

Subtract $1$ from $1$ .

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

Step 2.1.3.8.4

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.8.5

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Multiply $3$ by $- 1$ .

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

Step 2.3.5

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

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Step 2.3.5.1

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

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

Step 2.3.5.2

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

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

Step 2.3.5.3

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

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

Step 2.3.5.4

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

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

Step 2.3.5.5

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

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

Step 2.3.5.6

Multiply $- 1$ by $1$ .

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

Step 2.3.5.7

Subtract $1$ from $0$ .

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

Step 2.3.5.8

Multiply $- 3$ by $- 1$ .

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

Step 2.3.6

Subtract $3 x^{2}$ from $0$ .

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

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.3.10

Add $0$ and $\frac{d}{d x} [- x^{3}]$ .

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

Step 2.3.11

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

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

Step 2.3.12

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

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

Step 2.3.13

Multiply $3$ by $- 1$ .

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

Step 2.3.14

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

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

Step 2.3.15

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

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

Step 2.3.16

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 2.3.17

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

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

Step 2.3.18

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

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

Step 2.3.19

Multiply $- 1$ by $1$ .

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

Step 2.3.20

Move $- 1$ to the left of $1 - x^{3}$ .

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

Step 2.3.21

Rewrite $- 1 (1 - x^{3})$ as $- (1 - x^{3})$ .

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

Step 2.3.22

Simplify.

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Step 2.3.22.1

Apply the distributive property.

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

Step 2.3.22.2

Apply the distributive property.

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

Step 2.3.22.3

Combine terms.

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Step 2.3.22.3.1

Multiply $- 3$ by $1$ .

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

Step 2.3.22.3.2

Multiply $- 3$ by $- 1$ .

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

Step 2.3.22.3.3

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

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Step 2.3.22.3.3.1

Move $x^{2}$ .

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

Step 2.3.22.3.3.2

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

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Step 2.3.22.3.3.2.1

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

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

Step 2.3.22.3.3.2.2

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

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

Step 2.3.22.3.3.3

Add $2$ and $1$ .

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

Step 2.3.22.3.4

Multiply $- 1$ by $1$ .

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

Step 2.3.22.3.5

Multiply $- 1$ by $- 1$ .

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

Step 2.3.22.3.6

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

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

Step 2.3.22.3.7

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

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

Step 2.3.22.4

Reorder terms.

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

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 $1$ .

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

Step 3.1.2.2

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

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

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.

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

Step 3.1.2.3.1.2

Multiply $- 3$ by $1$ .

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

Step 3.1.2.3.2

Add $- 3$ and $3$ .

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

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

Factor $6$ out of $- 6$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$- 1$