Calculus Examples

$lim_{x \to \infty} \frac{3 x^{3} + 2 x^{2} + 8}{4 x^{3} + 6 x^{2}}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{3 x^{3}}{x^{3}} + \frac{2 x^{2}}{x^{3}} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

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 $x^{3}$ .

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Step 2.1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{3 x^{3}}{x^{3}} + \frac{2 x^{2}}{x^{3}} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.1.1.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{3 + \frac{2 x^{2}}{x^{3}} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.1.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.1.2.1

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

$lim_{x \to \infty} \frac{3 + \frac{x^{2} \cdot 2}{x^{3}} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

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

$lim_{x \to \infty} \frac{3 + \frac{x^{2} \cdot 2}{x^{2} x} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{3 + \frac{x^{2} \cdot 2}{x^{2} x} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

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.

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{\frac{4 x^{3}}{x^{3}} + \frac{6 x^{2}}{x^{3}}}$

Step 2.2.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{4 + \frac{6 x^{2}}{x^{3}}}$

Step 2.2.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.2.2.1

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

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{4 + \frac{x^{2} \cdot 6}{x^{3}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{4 + \frac{x^{2} \cdot 6}{x^{2} x}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{4 + \frac{x^{2} \cdot 6}{x^{2} x}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{8}{x^{3}}}{4 + \frac{6}{x}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 3 + \frac{2}{x} + \frac{8}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{8}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 2.5

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

$\frac{3 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{8}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 2.6

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

$\frac{3 + 2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{8}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 3

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

$\frac{3 + 2 \cdot 0 + lim_{x \to \infty} \frac{8}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 2 \cdot 0 + 8 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} 4 + \frac{6}{x}}$

Step 5

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

$\frac{3 + 2 \cdot 0 + 8 \cdot 0}{lim_{x \to \infty} 4 + \frac{6}{x}}$

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 $\infty$ .

$\frac{3 + 2 \cdot 0 + 8 \cdot 0}{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{6}{x}}$

Step 6.2

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

$\frac{3 + 2 \cdot 0 + 8 \cdot 0}{4 + lim_{x \to \infty} \frac{6}{x}}$

Step 6.3

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

$\frac{3 + 2 \cdot 0 + 8 \cdot 0}{4 + 6 lim_{x \to \infty} \frac{1}{x}}$

Step 7

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

$\frac{3 + 2 \cdot 0 + 8 \cdot 0}{4 + 6 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $2$ by $0$ .

$\frac{3 + 0 + 8 \cdot 0}{4 + 6 \cdot 0}$

Step 8.1.2

Multiply $8$ by $0$ .

$\frac{3 + 0 + 0}{4 + 6 \cdot 0}$

Step 8.1.3

Add $3 + 0$ and $0$ .

$\frac{3 + 0}{4 + 6 \cdot 0}$

Step 8.1.4

Add $3$ and $0$ .

$\frac{3}{4 + 6 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $6$ by $0$ .

$\frac{3}{4 + 0}$

Step 8.2.2

Add $4$ and $0$ .

$\frac{3}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$