Calculus Examples

$lim_{x \to \infty} \frac{2 x - x^{7}}{3 x^{7} - 2 x^{3} + 4 x + 1}$

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.

$\frac{lim_{x \to \infty} 2 x - x^{7}}{lim_{x \to \infty} 3 x^{7} - 2 x^{3} + 4 x + 1}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Reorder $2 x$ and $- x^{7}$ .

$\frac{lim_{x \to \infty} - x^{7} + 2 x}{lim_{x \to \infty} 3 x^{7} - 2 x^{3} + 4 x + 1}$

Step 1.2.2

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

$\frac{- \infty}{lim_{x \to \infty} 3 x^{7} - 2 x^{3} + 4 x + 1}$

Step 1.3

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

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

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.

$lim_{x \to \infty} \frac{2 x - x^{7}}{3 x^{7} - 2 x^{3} + 4 x + 1} = lim_{x \to \infty} \frac{\frac{d}{d x} [2 x - x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [2 x - x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.2

By the Sum Rule, the derivative of $2 x - x^{7}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{7}]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [2 x] + \frac{d}{d x} [- x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to \infty} \frac{2 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.3.2

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

$lim_{x \to \infty} \frac{2 \cdot 1 + \frac{d}{d x} [- x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.3.3

Multiply $2$ by $1$ .

$lim_{x \to \infty} \frac{2 + \frac{d}{d x} [- x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.4

Evaluate $\frac{d}{d x} [- x^{7}]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{7}$ with respect to $x$ is $- \frac{d}{d x} [x^{7}]$ .

$lim_{x \to \infty} \frac{2 - \frac{d}{d x} [x^{7}]}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$lim_{x \to \infty} \frac{2 - (7 x^{6})}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.4.3

Multiply $7$ by $- 1$ .

$lim_{x \to \infty} \frac{2 - 7 x^{6}}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.5

Reorder terms.

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{\frac{d}{d x} [3 x^{7} - 2 x^{3} + 4 x + 1]}$

Step 3.6

By the Sum Rule, the derivative of $3 x^{7} - 2 x^{3} + 4 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.7

Evaluate $\frac{d}{d x} [3 x^{7}]$ .

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Step 3.7.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{7}$ with respect to $x$ is $3 \frac{d}{d x} [x^{7}]$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{3 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{3 (7 x^{6}) + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.7.3

Multiply $7$ by $3$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} + \frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.8

Evaluate $\frac{d}{d x} [- 2 x^{3}]$ .

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Step 3.8.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.8.2

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

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 2 (3 x^{2}) + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.8.3

Multiply $3$ by $- 2$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [1]}$

Step 3.9

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

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Step 3.9.1

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

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + 4 \frac{d}{d x} [x] + \frac{d}{d x} [1]}$

Step 3.9.2

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

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + 4 \cdot 1 + \frac{d}{d x} [1]}$

Step 3.9.3

Multiply $4$ by $1$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + 4 + \frac{d}{d x} [1]}$

Step 3.10

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

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + 4 + 0}$

Step 3.11

Add $21 x^{6} - 6 x^{2} + 4$ and $0$ .

$lim_{x \to \infty} \frac{- 7 x^{6} + 2}{21 x^{6} - 6 x^{2} + 4}$

Step 4

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{6}$ .

$lim_{x \to \infty} \frac{\frac{- 7 x^{6}}{x^{6}} + \frac{2}{x^{6}}}{\frac{21 x^{6}}{x^{6}} + \frac{- 6 x^{2}}{x^{6}} + \frac{4}{x^{6}}}$

Step 5

Evaluate the limit.

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Step 5.1

Cancel the common factor of $x^{6}$ .

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Step 5.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{- 7 x^{6}}{x^{6}} + \frac{2}{x^{6}}}{\frac{21 x^{6}}{x^{6}} + \frac{- 6 x^{2}}{x^{6}} + \frac{4}{x^{6}}}$

Step 5.1.2

Divide $- 7$ by $1$ .

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{\frac{21 x^{6}}{x^{6}} + \frac{- 6 x^{2}}{x^{6}} + \frac{4}{x^{6}}}$

Step 5.2

Simplify each term.

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Step 5.2.1

Cancel the common factor of $x^{6}$ .

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Step 5.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{\frac{21 x^{6}}{x^{6}} + \frac{- 6 x^{2}}{x^{6}} + \frac{4}{x^{6}}}$

Step 5.2.1.2

Divide $21$ by $1$ .

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 + \frac{- 6 x^{2}}{x^{6}} + \frac{4}{x^{6}}}$

Step 5.2.2

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 5.2.2.1

Factor $x^{2}$ out of $- 6 x^{2}$ .

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 + \frac{x^{2} \cdot - 6}{x^{6}} + \frac{4}{x^{6}}}$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $x^{2}$ out of $x^{6}$ .

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 + \frac{x^{2} \cdot - 6}{x^{2} x^{4}} + \frac{4}{x^{6}}}$

Step 5.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 + \frac{x^{2} \cdot - 6}{x^{2} x^{4}} + \frac{4}{x^{6}}}$

Step 5.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 + \frac{- 6}{x^{4}} + \frac{4}{x^{6}}}$

Step 5.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{- 7 + \frac{2}{x^{6}}}{21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 5.3

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

$\frac{lim_{x \to \infty} - 7 + \frac{2}{x^{6}}}{lim_{x \to \infty} 21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 5.4

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

$\frac{- lim_{x \to \infty} 7 + lim_{x \to \infty} \frac{2}{x^{6}}}{lim_{x \to \infty} 21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 5.5

Evaluate the limit of $7$ which is constant as $x$ approaches $\infty$ .

$\frac{- 1 \cdot 7 + lim_{x \to \infty} \frac{2}{x^{6}}}{lim_{x \to \infty} 21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 5.6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 7 + 2 lim_{x \to \infty} \frac{1}{x^{6}}}{lim_{x \to \infty} 21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{6}}$ approaches $0$ .

$\frac{- 7 + 2 \cdot 0}{lim_{x \to \infty} 21 - \frac{6}{x^{4}} + \frac{4}{x^{6}}}$

Step 7

Evaluate the limit.

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Step 7.1

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

$\frac{- 7 + 2 \cdot 0}{lim_{x \to \infty} 21 - lim_{x \to \infty} \frac{6}{x^{4}} + lim_{x \to \infty} \frac{4}{x^{6}}}$

Step 7.2

Evaluate the limit of $21$ which is constant as $x$ approaches $\infty$ .

$\frac{- 7 + 2 \cdot 0}{21 - lim_{x \to \infty} \frac{6}{x^{4}} + lim_{x \to \infty} \frac{4}{x^{6}}}$

Step 7.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 7 + 2 \cdot 0}{21 - 6 lim_{x \to \infty} \frac{1}{x^{4}} + lim_{x \to \infty} \frac{4}{x^{6}}}$

Step 8

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

$\frac{- 7 + 2 \cdot 0}{21 - 6 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{6}}}$

Step 9

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 7 + 2 \cdot 0}{21 - 6 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{6}}}$

Step 10

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{6}}$ approaches $0$ .

$\frac{- 7 + 2 \cdot 0}{21 - 6 \cdot 0 + 4 \cdot 0}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $2$ by $0$ .

$\frac{- 7 + 0}{21 - 6 \cdot 0 + 4 \cdot 0}$

Step 11.1.2

Add $- 7$ and $0$ .

$\frac{- 7}{21 - 6 \cdot 0 + 4 \cdot 0}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 6$ by $0$ .

$\frac{- 7}{21 + 0 + 4 \cdot 0}$

Step 11.2.2

Multiply $4$ by $0$ .

$\frac{- 7}{21 + 0 + 0}$

Step 11.2.3

Add $21 + 0$ and $0$ .

$\frac{- 7}{21 + 0}$

Step 11.2.4

Add $21$ and $0$ .

$\frac{- 7}{21}$

Step 11.3

Cancel the common factor of $- 7$ and $21$ .

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Step 11.3.1

Factor $7$ out of $- 7$ .

$\frac{7 (- 1)}{21}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $7$ out of $21$ .

$\frac{7 \cdot - 1}{7 \cdot 3}$

Step 11.3.2.2

Cancel the common factor.

$\frac{7 \cdot - 1}{7 \cdot 3}$

Step 11.3.2.3

Rewrite the expression.

$\frac{- 1}{3}$

Step 11.4

Move the negative in front of the fraction.

$- \frac{1}{3}$