Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{4}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 2.1.1.1

Factor $x^{3}$ out of $3 x^{3}$ .

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

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 2.1.1.2.2

Cancel the common factor.

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

Step 2.1.1.2.3

Rewrite the expression.

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

Step 2.1.2

Cancel the common factor of $x$ and $x^{4}$ .

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Step 2.1.2.1

Factor $x$ out of $2 x$ .

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

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $x$ out of $x^{4}$ .

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

Step 2.1.2.2.2

Cancel the common factor.

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

Step 2.1.2.2.3

Rewrite the expression.

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

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{4}$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $2$ by $1$ .

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

Step 2.2.2

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 2.2.2.1

Factor $x^{3}$ out of $- 3 x^{3}$ .

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x^{3}$ out of $x^{4}$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Move the negative in front of the fraction.

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

Step 2.2.4

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

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

Step 6

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

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

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 $\infty$ .

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

Step 7.2

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

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

Step 7.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

$\frac{3 \cdot 0 + 2 \cdot 0 - 0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $3$ by $0$ .

$\frac{0 + 2 \cdot 0 - 0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11.1.2

Multiply $2$ by $0$ .

$\frac{0 + 0 - 0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11.1.3

Multiply $- 1$ by $0$ .

$\frac{0 + 0 + 0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11.1.4

Add $0 + 0$ and $0$ .

$\frac{0 + 0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11.1.5

Add $0$ and $0$ .

$\frac{0}{2 - 3 \cdot 0 - 2 \cdot 0}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $- 3$ by $0$ .

$\frac{0}{2 + 0 - 2 \cdot 0}$

Step 11.2.2

Multiply $- 2$ by $0$ .

$\frac{0}{2 + 0 + 0}$

Step 11.2.3

Add $2 + 0$ and $0$ .

$\frac{0}{2 + 0}$

Step 11.2.4

Add $2$ and $0$ .

$\frac{0}{2}$

Step 11.3

Divide $0$ by $2$ .

$0$