Calculus Examples

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

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.

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

Step 1.2

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

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Reorder $x$ and $- x^{3}$ .

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

Step 1.3.2

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

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

Step 1.3.3

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $7$ .

$lim_{x \to \infty} \frac{14 x}{\frac{d}{d x} [x - x^{3}]}$

Step 3.5

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

$lim_{x \to \infty} \frac{14 x}{\frac{d}{d x} [x] + \frac{d}{d x} [- x^{3}]}$

Step 3.6

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

$lim_{x \to \infty} \frac{14 x}{1 + \frac{d}{d x} [- x^{3}]}$

Step 3.7

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

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Step 3.7.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$lim_{x \to \infty} \frac{14 x}{1 - \frac{d}{d x} [x^{3}]}$

Step 3.7.2

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

$lim_{x \to \infty} \frac{14 x}{1 - (3 x^{2})}$

Step 3.7.3

Multiply $3$ by $- 1$ .

$lim_{x \to \infty} \frac{14 x}{1 - 3 x^{2}}$

Step 3.8

Reorder terms.

$lim_{x \to \infty} \frac{14 x}{- 3 x^{2} + 1}$

Step 4

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$14 lim_{x \to \infty} \frac{x}{- 3 x^{2} + 1}$

Step 5

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

$14 lim_{x \to \infty} \frac{\frac{x}{x^{2}}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6

Evaluate the limit.

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Step 6.1

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

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Step 6.1.1

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

$14 lim_{x \to \infty} \frac{\frac{x^{1}}{x^{2}}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.1.2

Factor $x$ out of $x^{1}$ .

$14 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x^{2}}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.1.3

Cancel the common factors.

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Step 6.1.3.1

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

$14 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x \cdot x}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.1.3.2

Cancel the common factor.

$14 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x \cdot x}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.1.3.3

Rewrite the expression.

$14 lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.2

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

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Step 6.2.1

Cancel the common factor.

$14 lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{- 3 x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

Step 6.2.2

Divide $- 3$ by $1$ .

$14 lim_{x \to \infty} \frac{\frac{1}{x}}{- 3 + \frac{1}{x^{2}}}$

Step 6.3

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

$14 \frac{lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} - 3 + \frac{1}{x^{2}}}$

Step 7

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

$14 \frac{0}{lim_{x \to \infty} - 3 + \frac{1}{x^{2}}}$

Step 8

Evaluate the limit.

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Step 8.1

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

$14 \frac{0}{- lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 8.2

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

$14 \frac{0}{- 1 \cdot 3 + lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 9

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

$14 \frac{0}{- 3 + 0}$

Step 10

Simplify the answer.

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Step 10.1

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

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Step 10.1.1

Factor $3$ out of $0$ .

$14 \frac{3 (0)}{- 3 + 0}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $3$ out of $- 3$ .

$14 \frac{3 \cdot 0}{3 \cdot - 1 + 0}$

Step 10.1.2.2

Factor $3$ out of $0$ .

$14 \frac{3 (0)}{3 \cdot - 1 + 3 \cdot 0}$

Step 10.1.2.3

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

$14 \frac{3 (0)}{3 \cdot (- 1 + 0)}$

Step 10.1.2.4

Cancel the common factor.

$14 \frac{3 \cdot 0}{3 \cdot (- 1 + 0)}$

Step 10.1.2.5

Rewrite the expression.

$14 \frac{0}{- 1 + 0}$

Step 10.2

Add $- 1$ and $0$ .

$14 (\frac{0}{- 1})$

Step 10.3

Divide $0$ by $- 1$ .

$14 \cdot 0$

Step 10.4

Multiply $14$ by $0$ .

$0$