Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Move the limit into the exponent.

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

Step 1.2.1.4

Evaluate the limit of $3$ which is constant as $x$ approaches $0$ .

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

Step 1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

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

Anything raised to $0$ is $1$ .

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

Step 1.2.3.1.2

Multiply $3$ by $1$ .

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

Step 1.2.3.1.3

Multiply $- 1$ by $3$ .

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

Step 1.2.3.2

Subtract $3$ from $3$ .

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

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 $0$ .

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

Step 1.3.2

Move the limit inside the logarithm.

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

Step 1.3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 1.3.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 1.3.5

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

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

Step 1.3.6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.3.6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.3.6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.3.7

Simplify the answer.

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Step 1.3.7.1

Simplify each term.

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Step 1.3.7.1.1

Add $1$ and $0$ .

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

Step 1.3.7.1.2

The natural logarithm of $1$ is $0$ .

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

Step 1.3.7.1.3

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

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

Step 1.3.7.1.4

Multiply $- 1$ by $0$ .

$\frac{0}{0 + 0}$

Step 1.3.7.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.3.7.3

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

Undefined

$\frac{0}{0}$

Step 1.3.8

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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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 $f (x) = \ln (x)$ and $g (x) = 1 - x$ .

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Step 3.7.1.1

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

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

Step 3.7.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 3.7.1.3

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

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

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

Step 3.7.6

Multiply $- 1$ by $1$ .

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

Step 3.7.7

Subtract $1$ from $0$ .

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

Step 3.7.8

Combine $\frac{1}{1 - x}$ and $- 1$ .

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

Step 3.7.9

Move the negative in front of the fraction.

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

Step 3.8

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

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Step 3.8.1

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

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

Step 3.8.2

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

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

Step 3.8.3

Multiply $3$ by $- 1$ .

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

Step 3.9

Simplify.

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Step 3.9.1

Combine terms.

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Step 3.9.1.1

To write $- 3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

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

Step 3.9.1.2

Combine $- 3 x^{2}$ and $\frac{1 - x}{1 - x}$ .

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

Step 3.9.1.3

Combine the numerators over the common denominator.

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

Step 3.9.2

Reorder terms.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

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Step 5.1

Combine $\frac{- x + 1}{- 3 x^{2} (1 - x) - 1}$ and $3$ .

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

Step 5.2

Combine $\frac{(- x + 1) \cdot 3}{- 3 x^{2} (1 - x) - 1}$ and $e^{x}$ .

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

Step 6

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

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

Step 7

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

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

Step 8

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 11

Move the limit into the exponent.

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

Step 12

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 13

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

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

Step 14

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 15

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

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

Step 16

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 17

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 18

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 19

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 19.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 19.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 19.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 19.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 20

Simplify the answer.

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Step 20.1

Simplify the numerator.

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Step 20.1.1

Add $0$ and $1$ .

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

Step 20.1.2

Multiply $e^{0}$ by $1$ .

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

Step 20.1.3

Anything raised to $0$ is $1$ .

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

Step 20.2

Simplify the denominator.

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Step 20.2.1

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

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

Step 20.2.2

Multiply $- 3$ by $0$ .

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

Step 20.2.3

Add $1$ and $0$ .

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

Step 20.2.4

Multiply $0$ by $1$ .

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

Step 20.2.5

Multiply $- 1$ by $1$ .

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

Step 20.2.6

Subtract $1$ from $0$ .

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

Step 20.3

Divide $1$ by $- 1$ .

$3 \cdot - 1$

Step 20.4

Multiply $3$ by $- 1$ .

$- 3$