Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(\frac{x - 3}{x})}^{x}$ as $e^{\ln ({(\frac{x - 3}{x})}^{x})}$ .

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

Step 2.2

Expand $\ln ({(\frac{x - 3}{x})}^{x})$ by moving $x$ outside the logarithm.

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

Step 3

Move the limit into the exponent.

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

Step 4

Rewrite $x \ln (\frac{x - 3}{x})$ as $\frac{\ln (\frac{x - 3}{x})}{\frac{1}{x}}$ .

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Move the limit inside the logarithm.

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

Step 5.1.2.2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

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

Step 5.1.2.3

Evaluate the limit.

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Step 5.1.2.3.1

Simplify each term.

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Step 5.1.2.3.1.1

Cancel the common factor of $x$ .

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Step 5.1.2.3.1.1.1

Cancel the common factor.

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

Step 5.1.2.3.1.1.2

Rewrite the expression.

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

Step 5.1.2.3.1.2

Move the negative in front of the fraction.

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

Step 5.1.2.3.2

Cancel the common factor of $x$ .

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Step 5.1.2.3.2.1

Cancel the common factor.

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

Step 5.1.2.3.2.2

Rewrite the expression.

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

Step 5.1.2.3.3

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

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

Step 5.1.2.3.4

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

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

Step 5.1.2.3.5

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

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

Step 5.1.2.3.6

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

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

Step 5.1.2.4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 5.1.2.5

Evaluate the limit.

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Step 5.1.2.5.1

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

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

Step 5.1.2.5.2

Simplify the answer.

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Step 5.1.2.5.2.1

Divide $1 - 3 \cdot 0$ by $1$ .

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

Step 5.1.2.5.2.2

Multiply $- 3$ by $0$ .

$e^{\frac{\ln (1 + 0)}{lim_{x \to \infty} \frac{1}{x}}}$

Step 5.1.2.5.2.3

Add $1$ and $0$ .

$e^{\frac{\ln (1)}{lim_{x \to \infty} \frac{1}{x}}}$

Step 5.1.2.5.2.4

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

$e^{\frac{0}{lim_{x \to \infty} \frac{1}{x}}}$

Step 5.1.3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$e^{\frac{0}{0}}$

Step 5.1.4

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

Undefined

$e^{\frac{0}{0}}$

Step 5.2

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.

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

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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 $g (x) = \frac{x - 3}{x}$ .

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Step 5.3.2.1

To apply the Chain Rule, set $u$ as $\frac{x - 3}{x}$ .

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Replace all occurrences of $u$ with $\frac{x - 3}{x}$ .

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

Step 5.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{x - 3}{x}$ .

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

Step 5.3.4

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

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

Step 5.3.5

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 3$ and $g (x) = x$ .

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

Step 5.3.6

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

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

Step 5.3.7

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

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

Step 5.3.8

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

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

Step 5.3.9

Add $1$ and $0$ .

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

Step 5.3.10

Multiply $x$ by $1$ .

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

Step 5.3.11

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

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

Step 5.3.12

Multiply $- 1$ by $1$ .

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

Step 5.3.13

Multiply $\frac{x}{x - 3}$ by $\frac{x - (x - 3)}{x^{2}}$ .

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

Step 5.3.14

Cancel the common factors.

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Step 5.3.14.1

Factor $x$ out of $(x - 3) x^{2}$ .

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

Step 5.3.14.2

Cancel the common factor.

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

Step 5.3.14.3

Rewrite the expression.

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

Step 5.3.15

Simplify.

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Step 5.3.15.1

Apply the distributive property.

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

Step 5.3.15.2

Apply the distributive property.

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

Step 5.3.15.3

Simplify the numerator.

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Step 5.3.15.3.1

Subtract $x$ from $x$ .

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

Step 5.3.15.3.2

Subtract $- 3$ from $0$ .

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

Step 5.3.15.3.3

Multiply $- 1$ by $- 3$ .

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

Step 5.3.15.4

Combine terms.

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Step 5.3.15.4.1

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

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

Step 5.3.15.4.2

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

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

Step 5.3.15.4.3

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

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

Step 5.3.15.4.4

Add $1$ and $1$ .

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

Step 5.3.15.5

Factor $x$ out of $x^{2} - 3 x$ .

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Step 5.3.15.5.1

Factor $x$ out of $x^{2}$ .

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

Step 5.3.15.5.2

Factor $x$ out of $- 3 x$ .

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

Step 5.3.15.5.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Step 5.3.16

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 5.3.17

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

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

Step 5.3.18

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5

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

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

Step 5.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.6.1

Factor $x$ out of $3 x^{2}$ .

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

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Cancel the common factor.

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

Step 5.6.2.2

Rewrite the expression.

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

Step 6

Evaluate the limit.

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Step 6.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 6.2

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

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

Step 7

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

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

Step 8

Evaluate the limit.

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Step 8.1

Cancel the common factor of $x$ .

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Step 8.1.1

Cancel the common factor.

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

Step 8.1.2

Rewrite the expression.

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

Step 8.2

Simplify each term.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Rewrite the expression.

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

Step 8.2.2

Move the negative in front of the fraction.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

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

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

Step 9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Multiply $- 3$ by $0$ .

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

Step 10.1.2

Add $1$ and $0$ .

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

Step 10.2

Cancel the common factor of $1$ .

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Step 10.2.1

Cancel the common factor.

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

Step 10.2.2

Rewrite the expression.

$e^{- 3 \cdot 1}$

Step 10.3

Multiply $- 3$ by $1$ .

$e^{- 3}$

Step 11

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{e^{3}}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e^{3}}$

Decimal Form:

$0.04978706 \dots$