Calculus Examples

$lim_{x \to \infty} \frac{x^{4}}{4^{x}}$

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.

$\frac{lim_{x \to \infty} x^{4}}{lim_{x \to \infty} 4^{x}}$

Step 1.1.2

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

$\frac{\infty}{lim_{x \to \infty} 4^{x}}$

Step 1.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $4^{x}$ 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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

$lim_{x \to \infty} \frac{4 x^{3}}{\frac{d}{d x} [4^{x}]}$

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

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

Step 2

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

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

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.

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

Step 3.1.2

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

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

Step 3.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $4^{x}$ approaches $\infty$ .

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

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.2

$lim_{x \to \infty} \frac{x^{3}}{4^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{3}]}{\frac{d}{d x} [4^{x}]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

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

Step 4

Move the term $\frac{3}{\ln (4)}$ outside of the limit because it is constant with respect to $x$ .

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

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.

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

Step 5.1.2

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

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

Step 5.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $4^{x}$ approaches $\infty$ .

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

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 5.2

$lim_{x \to \infty} \frac{x^{2}}{4^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [4^{x}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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

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

Step 5.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

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

Step 6

Move the term $\frac{2}{\ln (4)}$ outside of the limit because it is constant with respect to $x$ .

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

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.

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

Step 7.1.2

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

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

Step 7.1.3

Since the exponent $x$ approaches $\infty$ , the quantity $4^{x}$ approaches $\infty$ .

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

Step 7.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 7.2

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

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

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

Step 7.3.2

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

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

Step 7.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $4$ .

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

Step 8

Move the term $\frac{1}{\ln (4)}$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{4^{x}}$ approaches $0$ .

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

Step 10

Simplify the answer.

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Step 10.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 10.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

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

Step 10.3

Cancel the common factor of $4$ and $2$ .

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Step 10.3.1

Factor $2$ out of $4$ .

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

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $2$ out of $2 \ln (2)$ .

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

Step 10.3.2.2

Cancel the common factor.

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

Step 10.3.2.3

Rewrite the expression.

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

Step 10.4

Multiply $\frac{2}{\ln (2)} \cdot \frac{3}{\ln (4)}$ .

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Step 10.4.1

Multiply $\frac{2}{\ln (2)}$ by $\frac{3}{\ln (4)}$ .

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

Step 10.4.2

Multiply $2$ by $3$ .

$\frac{6}{\ln (2) \ln (4)} \cdot \frac{2}{\ln (4)} \frac{1}{\ln (4)} \cdot 0$

Step 10.5

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\frac{6}{\ln (2) \ln (2^{2})} \cdot \frac{2}{\ln (4)} \frac{1}{\ln (4)} \cdot 0$

Step 10.6

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$\frac{6}{\ln (2) (2 \ln (2))} \cdot \frac{2}{\ln (4)} \frac{1}{\ln (4)} \cdot 0$

Step 10.7

Cancel the common factor of $6$ and $2$ .

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Step 10.7.1

Factor $2$ out of $6$ .

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

Step 10.7.2

Cancel the common factors.

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Step 10.7.2.1

Factor $2$ out of $\ln (2) (2 \ln (2))$ .

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

Step 10.7.2.2

Cancel the common factor.

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

Step 10.7.2.3

Rewrite the expression.

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

Step 10.8

Simplify the denominator.

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Step 10.8.1

Raise $\ln (2)$ to the power of $1$ .

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

Step 10.8.2

Raise $\ln (2)$ to the power of $1$ .

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

Step 10.8.3

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

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

Step 10.8.4

Add $1$ and $1$ .

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

Step 10.9

Combine.

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

Step 10.10

Combine.

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

Step 10.11

Simplify the numerator.

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Step 10.11.1

Multiply $3$ by $2$ .

$\frac{6 \cdot 1}{\ln^{2} (2) \ln (4) \ln (4)} \cdot 0$

Step 10.11.2

Multiply $6$ by $1$ .

$\frac{6}{\ln^{2} (2) \ln (4) \ln (4)} \cdot 0$

Step 10.12

Simplify the denominator.

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Step 10.12.1

Raise $\ln (4)$ to the power of $1$ .

$\frac{6}{\ln^{2} (2) (\ln^{1} (4) \ln (4))} \cdot 0$

Step 10.12.2

Raise $\ln (4)$ to the power of $1$ .

$\frac{6}{\ln^{2} (2) (\ln^{1} (4) \ln^{1} (4))} \cdot 0$

Step 10.12.3

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

$\frac{6}{\ln^{2} (2) {\ln (4)}^{1 + 1}} \cdot 0$

Step 10.12.4

Add $1$ and $1$ .

$\frac{6}{\ln^{2} (2) \ln^{2} (4)} \cdot 0$

Step 10.13

Multiply $\frac{6}{\ln^{2} (2) \ln^{2} (4)}$ by $0$ .

$0$