Calculus Examples

$lim_{x \to \infty} \frac{6 x - \frac{1}{4 x}}{3 x + \frac{1}{x}}$

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.

$\frac{lim_{x \to \infty} 6 x - \frac{1}{4 x}}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

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 $\infty$ .

$\frac{lim_{x \to \infty} 6 x - lim_{x \to \infty} \frac{1}{4 x}}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.1.2

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

$\frac{\infty - lim_{x \to \infty} \frac{1}{4 x}}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.1.3

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

$\frac{\infty - \frac{1}{4} lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.2

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

$\frac{\infty - \frac{1}{4} \cdot 0}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Multiply $- \frac{1}{4} \cdot 0$ .

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Step 1.2.3.1.1

Multiply $0$ by $- 1$ .

$\frac{\infty + 0 (\frac{1}{4})}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.3.1.2

Multiply $0$ by $\frac{1}{4}$ .

$\frac{\infty + 0}{lim_{x \to \infty} 3 x + \frac{1}{x}}$

Step 1.2.3.2

Add $\infty$ and $0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

$\frac{\infty}{\infty + lim_{x \to \infty} \frac{1}{x}}$

Step 1.3.2

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

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

Step 1.3.3

Add $\infty$ and $0$ .

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

Step 1.3.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 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.

$lim_{x \to \infty} \frac{6 x - \frac{1}{4 x}}{3 x + \frac{1}{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [6 x - \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [6 x - \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.2

By the Sum Rule, the derivative of $6 x - \frac{1}{4 x}$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- \frac{1}{4 x}]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [6 x] + \frac{d}{d x} [- \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 3.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{6 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.3.2

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

$lim_{x \to \infty} \frac{6 \cdot 1 + \frac{d}{d x} [- \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.3.3

Multiply $6$ by $1$ .

$lim_{x \to \infty} \frac{6 + \frac{d}{d x} [- \frac{1}{4 x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{1}{4 x}]$ .

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Step 3.4.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 x}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]$ .

$lim_{x \to \infty} \frac{6 - \frac{1}{4} \frac{d}{d x} [\frac{1}{x}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.2

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

$lim_{x \to \infty} \frac{6 - \frac{1}{4} \frac{d}{d x} [x^{- 1}]}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.3

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

$lim_{x \to \infty} \frac{6 - \frac{1}{4} (- x^{- 2})}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.4

Multiply $- 1$ by $- 1$ .

$lim_{x \to \infty} \frac{6 + 1 (\frac{1}{4}) x^{- 2}}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.5

Multiply $\frac{1}{4}$ by $1$ .

$lim_{x \to \infty} \frac{6 + \frac{1}{4} x^{- 2}}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.6

Combine $\frac{1}{4}$ and $x^{- 2}$ .

$lim_{x \to \infty} \frac{6 + \frac{x^{- 2}}{4}}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.4.7

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to \infty} \frac{6 + \frac{1}{4 x^{2}}}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.5

Reorder terms.

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{\frac{d}{d x} [3 x + \frac{1}{x}]}$

Step 3.6

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{\frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{1}{x}]}$

Step 3.7

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

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Step 3.7.1

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}]}$

Step 3.7.2

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 \cdot 1 + \frac{d}{d x} [\frac{1}{x}]}$

Step 3.7.3

Multiply $3$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 + \frac{d}{d x} [\frac{1}{x}]}$

Step 3.8

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 3.8.1

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 + \frac{d}{d x} [x^{- 1}]}$

Step 3.8.2

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 - x^{- 2}}$

Step 3.9

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

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{3 - \frac{1}{x^{2}}}$

Step 3.10

Reorder terms.

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6}{- \frac{1}{x^{2}} + 3}$

Step 4

Combine terms.

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Step 4.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{4 x^{2}}{4 x^{2}}$ .

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + 6 \cdot \frac{4 x^{2}}{4 x^{2}}}{- \frac{1}{x^{2}} + 3}$

Step 4.2

Combine $6$ and $\frac{4 x^{2}}{4 x^{2}}$ .

$lim_{x \to \infty} \frac{\frac{1}{4 x^{2}} + \frac{6 (4 x^{2})}{4 x^{2}}}{- \frac{1}{x^{2}} + 3}$

Step 4.3

Combine the numerators over the common denominator.

$lim_{x \to \infty} \frac{\frac{1 + 6 (4 x^{2})}{4 x^{2}}}{- \frac{1}{x^{2}} + 3}$

Step 4.4

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

$lim_{x \to \infty} \frac{\frac{1 + 6 (4 x^{2})}{4 x^{2}}}{- \frac{1}{x^{2}} + \frac{3 x^{2}}{x^{2}}}$

Step 4.5

Combine the numerators over the common denominator.

$lim_{x \to \infty} \frac{\frac{1 + 6 (4 x^{2})}{4 x^{2}}}{\frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{lim_{x \to \infty} \frac{1 + 6 (4 x^{2})}{4 x^{2}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 5.2

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

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{1 + 6 (4 x^{2})}{x^{2}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 5.3

Multiply $4$ by $6$ .

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{1 + 24 x^{2}}{x^{2}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 6

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{\frac{1}{x^{2}} + \frac{24 x^{2}}{x^{2}}}{\frac{x^{2}}{x^{2}}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7

Evaluate the limit.

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Step 7.1

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

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Step 7.1.1

Cancel the common factor.

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{\frac{1}{x^{2}} + \frac{24 x^{2}}{x^{2}}}{\frac{x^{2}}{x^{2}}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7.1.2

Divide $24$ by $1$ .

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{\frac{1}{x^{2}} + 24}{\frac{x^{2}}{x^{2}}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7.2

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

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Step 7.2.1

Cancel the common factor.

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{\frac{1}{x^{2}} + 24}{\frac{x^{2}}{x^{2}}}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7.2.2

Rewrite the expression.

$\frac{\frac{1}{4} lim_{x \to \infty} \frac{\frac{1}{x^{2}} + 24}{1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7.3

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

$\frac{\frac{1}{4} \cdot \frac{lim_{x \to \infty} \frac{1}{x^{2}} + 24}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 7.4

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

$\frac{\frac{1}{4} \cdot \frac{lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 24}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{\frac{1}{4} \cdot \frac{0 + lim_{x \to \infty} 24}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 9

Evaluate the limit.

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Step 9.1

Evaluate the limit of $24$ which is constant as $x$ approaches $\infty$ .

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{lim_{x \to \infty} 1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 9.2

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

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- 1 + 3 x^{2}}{x^{2}}}$

Step 10

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- \frac{1}{x^{2}} + \frac{3 x^{2}}{x^{2}}}{\frac{x^{2}}{x^{2}}}}$

Step 11

Evaluate the limit.

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Step 11.1

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

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Step 11.1.1

Cancel the common factor.

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- \frac{1}{x^{2}} + \frac{3 x^{2}}{x^{2}}}{\frac{x^{2}}{x^{2}}}}$

Step 11.1.2

Divide $3$ by $1$ .

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- \frac{1}{x^{2}} + 3}{\frac{x^{2}}{x^{2}}}}$

Step 11.2

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

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Step 11.2.1

Cancel the common factor.

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- \frac{1}{x^{2}} + 3}{\frac{x^{2}}{x^{2}}}}$

Step 11.2.2

Rewrite the expression.

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{lim_{x \to \infty} \frac{- \frac{1}{x^{2}} + 3}{1}}$

Step 11.3

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

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{\frac{lim_{x \to \infty} - \frac{1}{x^{2}} + 3}{lim_{x \to \infty} 1}}$

Step 11.4

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

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{\frac{- lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} 3}{lim_{x \to \infty} 1}}$

Step 12

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{\frac{- 0 + lim_{x \to \infty} 3}{lim_{x \to \infty} 1}}$

Step 13

Evaluate the limit.

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Step 13.1

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

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{\frac{0 + 3}{lim_{x \to \infty} 1}}$

Step 13.2

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

$\frac{\frac{1}{4} \cdot \frac{0 + 24}{1}}{\frac{0 + 3}{1}}$

Step 13.3

Simplify the answer.

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Step 13.3.1

Divide $0 + 24$ by $1$ .

$\frac{\frac{1}{4} (0 + 24)}{\frac{0 + 3}{1}}$

Step 13.3.2

Divide $0 + 3$ by $1$ .

$\frac{\frac{1}{4} (0 + 24)}{0 + 3}$

Step 13.3.3

Cancel the common factor of $0 + 24$ and $0 + 3$ .

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Step 13.3.3.1

Factor $3$ out of $\frac{1}{4} (0 + 24)$ .

$\frac{3 (\frac{1}{4} (0 + 8))}{0 + 3}$

Step 13.3.3.2

Cancel the common factors.

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Step 13.3.3.2.1

Factor $3$ out of $0$ .

$\frac{3 (\frac{1}{4} (0 + 8))}{3 (0) + 3}$

Step 13.3.3.2.2

Factor $3$ out of $3$ .

$\frac{3 (\frac{1}{4} (0 + 8))}{3 \cdot 0 + 3 \cdot 1}$

Step 13.3.3.2.3

Factor $3$ out of $3 \cdot 0 + 3 \cdot 1$ .

$\frac{3 (\frac{1}{4} (0 + 8))}{3 \cdot (0 + 1)}$

Step 13.3.3.2.4

Cancel the common factor.

$\frac{3 (\frac{1}{4} (0 + 8))}{3 \cdot (0 + 1)}$

Step 13.3.3.2.5

Rewrite the expression.

$\frac{\frac{1}{4} (0 + 8)}{0 + 1}$

Step 13.3.4

Add $0$ and $8$ .

$\frac{\frac{1}{4} \cdot 8}{0 + 1}$

Step 13.3.5

Add $0$ and $1$ .

$\frac{\frac{1}{4} \cdot 8}{1}$

Step 13.3.6

Combine $\frac{1}{4}$ and $8$ .

$\frac{\frac{8}{4}}{1}$

Step 13.3.7

Divide $8$ by $4$ .

$\frac{2}{1}$

Step 13.3.8

Divide $2$ by $1$ .

$2$