Calculus Examples

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

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.

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

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 $- 1$ .

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

Step 1.2.1.2

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

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

Step 1.2.1.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Multiply $2$ by $- 1$ .

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

Step 1.2.3.2

Subtract $2$ from $2$ .

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

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 $- 1$ .

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

Step 1.3.2

Move the limit into the exponent.

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

Step 1.3.3

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

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

Step 1.3.4

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

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

Step 1.3.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

$\frac{0}{e^{- 2 - 2 \cdot - 1} + {(- 1)}^{3}}$

Step 1.3.8

Simplify the answer.

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Step 1.3.8.1

Simplify each term.

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Step 1.3.8.1.1

Multiply $- 2$ by $- 1$ .

$\frac{0}{e^{- 2 + 2} + {(- 1)}^{3}}$

Step 1.3.8.1.2

Add $- 2$ and $2$ .

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

Step 1.3.8.1.3

Anything raised to $0$ is $1$ .

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

Step 1.3.8.1.4

Raise $- 1$ to the power of $3$ .

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

Step 1.3.8.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.8.3

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

Undefined

$\frac{0}{0}$

Step 1.3.9

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $2$ by $1$ .

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

Step 3.5

Add $0$ and $2$ .

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

Step 3.6

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

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

Step 3.7

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

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Step 3.7.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 2 - 2 x$ .

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Step 3.7.1.1

To apply the Chain Rule, set $u$ as $- 2 - 2 x$ .

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

Step 3.7.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 3.7.1.3

Replace all occurrences of $u$ with $- 2 - 2 x$ .

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

Step 3.7.2

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

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

Step 3.7.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 3.7.4

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

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

Step 3.7.5

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

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

Step 3.7.6

Multiply $- 2$ by $1$ .

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

Step 3.7.7

Subtract $2$ from $0$ .

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

Step 3.7.8

Move $- 2$ to the left of $e^{- 2 - 2 x}$ .

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

Step 3.8

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

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

Step 3.9

Reorder terms.

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 11

Move the limit into the exponent.

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

Step 12

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

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

Step 13

Evaluate the limit of $2$ which is constant as $x$ approaches $- 1$ .

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

Step 14

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

$2 \frac{1}{3 {(- 1)}^{2} - 2 e^{- 2 - 2 \cdot - 1}}$

Step 16

Simplify the answer.

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Step 16.1

Simplify the denominator.

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Step 16.1.1

Raise $- 1$ to the power of $2$ .

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

Step 16.1.2

Multiply $3$ by $1$ .

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

Step 16.1.3

Multiply $- 2$ by $- 1$ .

$2 \frac{1}{3 - 2 e^{- 2 + 2}}$

Step 16.1.4

Add $- 2$ and $2$ .

$2 \frac{1}{3 - 2 e^{0}}$

Step 16.1.5

Anything raised to $0$ is $1$ .

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

Step 16.1.6

Multiply $- 2$ by $1$ .

$2 \frac{1}{3 - 2}$

Step 16.1.7

Subtract $2$ from $3$ .

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

Step 16.2

Cancel the common factor of $1$ .

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Step 16.2.1

Cancel the common factor.

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

Step 16.2.2

Rewrite the expression.

$2 \cdot 1$

Step 16.3

Multiply $2$ by $1$ .

$2$