Calculus Examples

$lim_{x \to \infty} \frac{x^{5}}{5^{x}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to \infty} x^{5}}{lim_{x \to \infty} 5^{x}}$

Step 1.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{\infty}{lim_{x \to \infty} 5^{x}}$

Step 1.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $5^{x}$ approaches $\infty$ .

$\frac{\infty}{\infty}$

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{\infty}$

Step 1.2

Since $\frac{\infty}{\infty}$ 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.

$lim_{x \to \infty} \frac{x^{5}}{5^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{5}]}{\frac{d}{d x} [5^{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [x^{5}]}{\frac{d}{d x} [5^{x}]}$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$lim_{x \to \infty} \frac{5 x^{4}}{\frac{d}{d x} [5^{x}]}$

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$lim_{x \to \infty} \frac{5 x^{4}}{5^{x} \ln (5)}$

Step 2

Move the term $\frac{5}{\ln (5)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{\ln (5)} lim_{x \to \infty} \frac{x^{4}}{5^{x}}$

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.

$\frac{5}{\ln (5)} \cdot \frac{lim_{x \to \infty} x^{4}}{lim_{x \to \infty} 5^{x}}$

Step 3.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{5}{\ln (5)} \cdot \frac{\infty}{lim_{x \to \infty} 5^{x}}$

Step 3.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $5^{x}$ approaches $\infty$ .

$\frac{5}{\ln (5)} \cdot \frac{\infty}{\infty}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{5}{\ln (5)} \cdot \frac{\infty}{\infty}$

Step 3.2

$lim_{x \to \infty} \frac{x^{4}}{5^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{4}]}{\frac{d}{d x} [5^{x}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{5}{\ln (5)} lim_{x \to \infty} \frac{\frac{d}{d x} [x^{4}]}{\frac{d}{d x} [5^{x}]}$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{5}{\ln (5)} lim_{x \to \infty} \frac{4 x^{3}}{\frac{d}{d x} [5^{x}]}$

Step 3.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\frac{5}{\ln (5)} lim_{x \to \infty} \frac{4 x^{3}}{5^{x} \ln (5)}$

Step 4

Move the term $\frac{4}{\ln (5)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} lim_{x \to \infty} \frac{x^{3}}{5^{x}}$

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.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{lim_{x \to \infty} x^{3}}{lim_{x \to \infty} 5^{x}}$

Step 5.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{\infty}{lim_{x \to \infty} 5^{x}}$

Step 5.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $5^{x}$ approaches $\infty$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{\infty}{\infty}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{\infty}{\infty}$

Step 5.2

$lim_{x \to \infty} \frac{x^{3}}{5^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5^{x}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [5^{x}]}$

Step 5.3.2

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

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} lim_{x \to \infty} \frac{3 x^{2}}{\frac{d}{d x} [5^{x}]}$

Step 5.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} lim_{x \to \infty} \frac{3 x^{2}}{5^{x} \ln (5)}$

Step 6

Move the term $\frac{3}{\ln (5)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} lim_{x \to \infty} \frac{x^{2}}{5^{x}}$

Step 7

Apply L'Hospital's rule.

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Step 7.1

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

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Step 7.1.1

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

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{lim_{x \to \infty} x^{2}}{lim_{x \to \infty} 5^{x}}$

Step 7.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{\infty}{lim_{x \to \infty} 5^{x}}$

Step 7.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $5^{x}$ approaches $\infty$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{\infty}{\infty}$

Step 7.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{\infty}{\infty}$

Step 7.2

$lim_{x \to \infty} \frac{x^{2}}{5^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [5^{x}]}$

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [5^{x}]}$

Step 7.3.2

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

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} lim_{x \to \infty} \frac{2 x}{\frac{d}{d x} [5^{x}]}$

Step 7.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} lim_{x \to \infty} \frac{2 x}{5^{x} \ln (5)}$

Step 8

Move the term $\frac{2}{\ln (5)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} lim_{x \to \infty} \frac{x}{5^{x}}$

Step 9

Apply L'Hospital's rule.

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Step 9.1

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

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Step 9.1.1

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

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{lim_{x \to \infty} x}{lim_{x \to \infty} 5^{x}}$

Step 9.1.2

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{\infty}{lim_{x \to \infty} 5^{x}}$

Step 9.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $5^{x}$ approaches $\infty$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{\infty}{\infty}$

Step 9.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{\infty}{\infty}$

Step 9.2

$lim_{x \to \infty} \frac{x}{5^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [5^{x}]}$

Step 9.3

Find the derivative of the numerator and denominator.

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Step 9.3.1

Differentiate the numerator and denominator.

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} lim_{x \to \infty} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [5^{x}]}$

Step 9.3.2

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

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} lim_{x \to \infty} \frac{1}{\frac{d}{d x} [5^{x}]}$

Step 9.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} lim_{x \to \infty} \frac{1}{5^{x} \ln (5)}$

Step 10

Move the term $\frac{1}{\ln (5)}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} lim_{x \to \infty} \frac{1}{5^{x}}$

Step 11

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{5^{x}}$ approaches $0$ .

$\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)} \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12

Simplify the answer.

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Step 12.1

Multiply $\frac{5}{\ln (5)} \cdot \frac{4}{\ln (5)}$ .

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Step 12.1.1

Multiply $\frac{5}{\ln (5)}$ by $\frac{4}{\ln (5)}$ .

$\frac{5 \cdot 4}{\ln (5) \ln (5)} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.1.2

Multiply $5$ by $4$ .

$\frac{20}{\ln (5) \ln (5)} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.1.3

Raise $\ln (5)$ to the power of $1$ .

$\frac{20}{\ln^{1} (5) \ln (5)} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.1.4

Raise $\ln (5)$ to the power of $1$ .

$\frac{20}{\ln^{1} (5) \ln^{1} (5)} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.1.5

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

$\frac{20}{{\ln (5)}^{1 + 1}} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.1.6

Add $1$ and $1$ .

$\frac{20}{\ln^{2} (5)} \cdot \frac{3}{\ln (5)} \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.2

Combine.

$\frac{20 \cdot 3}{\ln^{2} (5) \ln (5)} \cdot \frac{2}{\ln (5)} \frac{1}{\ln (5)} \cdot 0$

Step 12.3

Combine.

$\frac{20 \cdot 3 \cdot 2}{\ln^{2} (5) \ln (5) \ln (5)} \cdot \frac{1}{\ln (5)} \cdot 0$

Step 12.4

Combine.

$\frac{20 \cdot 3 \cdot 2 \cdot 1}{\ln^{2} (5) \ln (5) \ln (5) \ln (5)} \cdot 0$

Step 12.5

Simplify the numerator.

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Step 12.5.1

Multiply $20$ by $3$ .

$\frac{60 \cdot 2 \cdot 1}{\ln^{2} (5) \ln (5) \ln (5) \ln (5)} \cdot 0$

Step 12.5.2

Multiply $60$ by $2$ .

$\frac{120 \cdot 1}{\ln^{2} (5) \ln (5) \ln (5) \ln (5)} \cdot 0$

Step 12.5.3

Multiply $120$ by $1$ .

$\frac{120}{\ln^{2} (5) \ln (5) \ln (5) \ln (5)} \cdot 0$

Step 12.6

Simplify the denominator.

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Step 12.6.1

Raise $\ln (5)$ to the power of $1$ .

$\frac{120}{\ln^{2} (5) \ln^{1} (5) \ln (5) \ln (5)} \cdot 0$

Step 12.6.2

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

$\frac{120}{{\ln (5)}^{2 + 1} \ln (5) \ln (5)} \cdot 0$

Step 12.6.3

Add $2$ and $1$ .

$\frac{120}{\ln^{3} (5) \ln (5) \ln (5)} \cdot 0$

Step 12.6.4

Raise $\ln (5)$ to the power of $1$ .

$\frac{120}{\ln^{3} (5) \ln^{1} (5) \ln (5)} \cdot 0$

Step 12.6.5

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

$\frac{120}{{\ln (5)}^{3 + 1} \ln (5)} \cdot 0$

Step 12.6.6

Add $3$ and $1$ .

$\frac{120}{\ln^{4} (5) \ln (5)} \cdot 0$

Step 12.6.7

Multiply $\ln^{4} (5)$ by $\ln (5)$ by adding the exponents.

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Step 12.6.7.1

Multiply $\ln^{4} (5)$ by $\ln (5)$ .

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Step 12.6.7.1.1

Raise $\ln (5)$ to the power of $1$ .

$\frac{120}{\ln^{4} (5) \ln^{1} (5)} \cdot 0$

Step 12.6.7.1.2

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

$\frac{120}{{\ln (5)}^{4 + 1}} \cdot 0$

Step 12.6.7.2

Add $4$ and $1$ .

$\frac{120}{\ln^{5} (5)} \cdot 0$

Step 12.7

Multiply $\frac{120}{\ln^{5} (5)}$ by $0$ .

$0$