Calculus Examples

$lim_{x \to \infty} \frac{3 x^{3} + 5 x}{4 x^{4} + 10 x^{3} + 2}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{3 x^{3}}{x^{4}} + \frac{5 x}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

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 $3 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 3}{x^{4}} + \frac{5 x}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

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

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 3}{x^{3} x} + \frac{5 x}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{3} \cdot 3}{x^{3} x} + \frac{5 x}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5 x}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

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 $5 x$ .

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 5}{x^{4}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 5}{x \cdot x^{3}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 5}{x \cdot x^{3}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

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.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{\frac{4 x^{4}}{x^{4}} + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.2.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{4 + \frac{10 x^{3}}{x^{4}} + \frac{2}{x^{4}}}$

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 $10 x^{3}$ .

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{4 + \frac{x^{3} \cdot 10}{x^{4}} + \frac{2}{x^{4}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{4 + \frac{x^{3} \cdot 10}{x^{3} x} + \frac{2}{x^{4}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{4 + \frac{x^{3} \cdot 10}{x^{3} x} + \frac{2}{x^{4}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{5}{x^{3}}}{4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{3}{x} + \frac{5}{x^{3}}}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{3}{x} + lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 2.5

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

$\frac{3 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 3

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

$\frac{3 \cdot 0 + lim_{x \to \infty} \frac{5}{x^{3}}}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 \cdot 0 + 5 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 5

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

$\frac{3 \cdot 0 + 5 \cdot 0}{lim_{x \to \infty} 4 + \frac{10}{x} + \frac{2}{x^{4}}}$

Step 6

Evaluate the limit.

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Step 6.1

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

$\frac{3 \cdot 0 + 5 \cdot 0}{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{10}{x} + lim_{x \to \infty} \frac{2}{x^{4}}}$

Step 6.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{3 \cdot 0 + 5 \cdot 0}{4 + lim_{x \to \infty} \frac{10}{x} + lim_{x \to \infty} \frac{2}{x^{4}}}$

Step 6.3

Move the term $10$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 \cdot 0 + 5 \cdot 0}{4 + 10 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{2}{x^{4}}}$

Step 7

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

$\frac{3 \cdot 0 + 5 \cdot 0}{4 + 10 \cdot 0 + lim_{x \to \infty} \frac{2}{x^{4}}}$

Step 8

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

$\frac{3 \cdot 0 + 5 \cdot 0}{4 + 10 \cdot 0 + 2 lim_{x \to \infty} \frac{1}{x^{4}}}$

Step 9

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

$\frac{3 \cdot 0 + 5 \cdot 0}{4 + 10 \cdot 0 + 2 \cdot 0}$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of $3 \cdot 0 + 5 \cdot 0$ and $4 + 10 \cdot 0 + 2 \cdot 0$ .

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Step 10.1.1

Reorder terms.

$\frac{0 \cdot 3 + 0 \cdot 5}{4 + 10 \cdot 0 + 2 \cdot 0}$

Step 10.1.2

Factor $2$ out of $0 \cdot 3$ .

$\frac{2 (0 \cdot 3) + 0 \cdot 5}{4 + 10 \cdot 0 + 2 \cdot 0}$

Step 10.1.3

Factor $2$ out of $0 \cdot 5$ .

$\frac{2 (0 \cdot 3) + 2 (0 \cdot 5)}{4 + 10 \cdot 0 + 2 \cdot 0}$

Step 10.1.4

Factor $2$ out of $2 (0 \cdot 3) + 2 (0 \cdot 5)$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{4 + 10 \cdot 0 + 2 \cdot 0}$

Step 10.1.5

Cancel the common factors.

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Step 10.1.5.1

Factor $2$ out of $4$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2) + 10 \cdot 0 + 2 \cdot 0}$

Step 10.1.5.2

Factor $2$ out of $10 \cdot 0$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2) + 2 (5 \cdot 0) + 2 \cdot 0}$

Step 10.1.5.3

Factor $2$ out of $2 (2) + 2 (5 \cdot 0)$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2 + 5 \cdot 0) + 2 \cdot 0}$

Step 10.1.5.4

Factor $2$ out of $2 \cdot 0$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2 + 5 \cdot 0) + 2 (0)}$

Step 10.1.5.5

Factor $2$ out of $2 (2 + 5 \cdot 0) + 2 (0)$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2 + 5 \cdot 0 + 0)}$

Step 10.1.5.6

Cancel the common factor.

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (2 + 5 \cdot 0 + 0)}$

Step 10.1.5.7

Rewrite the expression.

$\frac{0 \cdot 3 + 0 \cdot 5}{2 + 5 \cdot 0 + 0}$

Step 10.2

Cancel the common factor of $0 \cdot 3 + 0 \cdot 5$ and $2 + 5 \cdot 0 + 0$ .

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Step 10.2.1

Reorder terms.

$\frac{0 \cdot 3 + 0 \cdot 5}{2 + 0 \cdot 5 + 0}$

Step 10.2.2

Factor $2$ out of $0 \cdot 3$ .

$\frac{2 (0 \cdot 3) + 0 \cdot 5}{2 + 0 \cdot 5 + 0}$

Step 10.2.3

Factor $2$ out of $0 \cdot 5$ .

$\frac{2 (0 \cdot 3) + 2 (0 \cdot 5)}{2 + 0 \cdot 5 + 0}$

Step 10.2.4

Factor $2$ out of $2 (0 \cdot 3) + 2 (0 \cdot 5)$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 + 0 \cdot 5 + 0}$

Step 10.2.5

Cancel the common factors.

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Step 10.2.5.1

Factor $2$ out of $2$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1) + 0 \cdot 5 + 0}$

Step 10.2.5.2

Factor $2$ out of $0 \cdot 5$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1) + 2 (0 \cdot 5) + 0}$

Step 10.2.5.3

Factor $2$ out of $2 (1) + 2 (0 \cdot 5)$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1 + 0 \cdot 5) + 0}$

Step 10.2.5.4

Factor $2$ out of $0$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1 + 0 \cdot 5) + 2 \cdot 0}$

Step 10.2.5.5

Factor $2$ out of $2 (1 + 0 \cdot 5) + 2 \cdot 0$ .

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1 + 0 \cdot 5 + 0)}$

Step 10.2.5.6

Cancel the common factor.

$\frac{2 (0 \cdot 3 + 0 \cdot 5)}{2 (1 + 0 \cdot 5 + 0)}$

Step 10.2.5.7

Rewrite the expression.

$\frac{0 \cdot 3 + 0 \cdot 5}{1 + 0 \cdot 5 + 0}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Multiply $0$ by $3$ .

$\frac{0 + 0 \cdot 5}{1 + 0 \cdot 5 + 0}$

Step 10.3.2

Multiply $0$ by $5$ .

$\frac{0 + 0}{1 + 0 \cdot 5 + 0}$

Step 10.3.3

Add $0$ and $0$ .

$\frac{0}{1 + 0 \cdot 5 + 0}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Multiply $0$ by $5$ .

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

Step 10.4.2

Add $1 + 0$ and $0$ .

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

Step 10.4.3

Add $1$ and $0$ .

$\frac{0}{1}$

Step 10.5

Divide $0$ by $1$ .

$0$