Calculus Examples

$lim_{x \to \infty} \frac{3 x^{2} + 4 x + 3}{x^{3} + x + 14}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{3 x^{2}}{x^{3}} + \frac{4 x}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

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 $x^{2}$ and $x^{3}$ .

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Step 2.1.1.1

Factor $x^{2}$ out of $3 x^{2}$ .

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 3}{x^{3}} + \frac{4 x}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 3}{x^{2} x} + \frac{4 x}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x^{2} \cdot 3}{x^{2} x} + \frac{4 x}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4 x}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.2

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

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Step 2.1.2.1

Factor $x$ out of $4 x$ .

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 4}{x^{3}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 4}{x \cdot x^{2}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{x \cdot 4}{x \cdot x^{2}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.2.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{x}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.2.2

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

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Step 2.2.2.1

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{x^{1}}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.2.2.2

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{x \cdot 1}{x^{3}} + \frac{14}{x^{3}}}$

Step 2.2.2.3

Cancel the common factors.

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Step 2.2.2.3.1

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

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{x \cdot 1}{x \cdot x^{2}} + \frac{14}{x^{3}}}$

Step 2.2.2.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{x \cdot 1}{x \cdot x^{2}} + \frac{14}{x^{3}}}$

Step 2.2.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{3}{x} + \frac{4}{x^{2}} + \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{3}{x} + lim_{x \to \infty} \frac{4}{x^{2}} + lim_{x \to \infty} \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 2.5

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

$\frac{3 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{4}{x^{2}} + lim_{x \to \infty} \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 3

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

$\frac{3 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{2}} + lim_{x \to \infty} \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 4

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

$\frac{3 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 5

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

$\frac{3 \cdot 0 + 4 \cdot 0 + lim_{x \to \infty} \frac{3}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 6

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

Step 7

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{lim_{x \to \infty} 1 + \frac{1}{x^{2}} + \frac{14}{x^{3}}}$

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 $\infty$ .

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{14}{x^{3}}}$

Step 8.2

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{1 + lim_{x \to \infty} \frac{1}{x^{2}} + lim_{x \to \infty} \frac{14}{x^{3}}}$

Step 9

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{1 + 0 + lim_{x \to \infty} \frac{14}{x^{3}}}$

Step 10

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{1 + 0 + 14 lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 11

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

$\frac{3 \cdot 0 + 4 \cdot 0 + 3 \cdot 0}{1 + 0 + 14 \cdot 0}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $3$ by $0$ .

$\frac{0 + 4 \cdot 0 + 3 \cdot 0}{1 + 0 + 14 \cdot 0}$

Step 12.1.2

Multiply $4$ by $0$ .

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

Step 12.1.3

Multiply $3$ by $0$ .

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

Step 12.1.4

Add $0 + 0$ and $0$ .

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

Step 12.1.5

Add $0$ and $0$ .

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

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $14$ by $0$ .

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

Step 12.2.2

Add $1 + 0$ and $0$ .

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

Step 12.2.3

Add $1$ and $0$ .

$\frac{0}{1}$

Step 12.3

Divide $0$ by $1$ .

$0$