Calculus Examples

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

Step 1

Move the exponent $3$ from ${(\frac{1 + 3 x}{1 + 4 x^{2} + 3 x^{4}})}^{3}$ outside the limit using the Limits Power Rule.

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

Step 2

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

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

Step 3

Evaluate the limit.

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Step 3.1

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

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Step 3.1.1

Factor $x$ out of $3 x$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 3.2.1.1

Factor $x^{2}$ out of $4 x^{2}$ .

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

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 3.2.1.2.2

Cancel the common factor.

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

Step 3.2.1.2.3

Rewrite the expression.

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

Step 3.2.2

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

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $3$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

${(\frac{0 + 3 \cdot 0}{0 + 4 \cdot 0 + lim_{x \to \infty} 3})}^{3}$

Step 11

Evaluate the limit.

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Step 11.1

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

${(\frac{0 + 3 \cdot 0}{0 + 4 \cdot 0 + 3})}^{3}$

Step 11.2

Simplify the answer.

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Step 11.2.1

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

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Step 11.2.1.1

Reorder terms.

${(\frac{0 + 3 \cdot 0}{0 + 0 \cdot 4 + 3})}^{3}$

Step 11.2.1.2

Factor $3$ out of $0$ .

${(\frac{3 (0) + 3 \cdot 0}{0 + 0 \cdot 4 + 3})}^{3}$

Step 11.2.1.3

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

${(\frac{3 (0) + 3 (0)}{0 + 0 \cdot 4 + 3})}^{3}$

Step 11.2.1.4

Factor $3$ out of $3 (0) + 3 (0)$ .

${(\frac{3 (0 + 0)}{0 + 0 \cdot 4 + 3})}^{3}$

Step 11.2.1.5

Cancel the common factors.

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Step 11.2.1.5.1

Factor $3$ out of $0$ .

${(\frac{3 (0 + 0)}{3 \cdot 0 + 0 \cdot 4 + 3})}^{3}$

Step 11.2.1.5.2

Factor $3$ out of $0 \cdot 4$ .

${(\frac{3 (0 + 0)}{3 \cdot 0 + 3 (0 \cdot 4) + 3})}^{3}$

Step 11.2.1.5.3

Factor $3$ out of $3 (0) + 3 (0 \cdot 4)$ .

${(\frac{3 (0 + 0)}{3 (0 + 0 \cdot 4) + 3})}^{3}$

Step 11.2.1.5.4

Factor $3$ out of $3$ .

${(\frac{3 (0 + 0)}{3 (0 + 0 \cdot 4) + 3 \cdot 1})}^{3}$

Step 11.2.1.5.5

Factor $3$ out of $3 (0 + 0 \cdot 4) + 3 \cdot 1$ .

${(\frac{3 (0 + 0)}{3 (0 + 0 \cdot 4 + 1)})}^{3}$

Step 11.2.1.5.6

Cancel the common factor.

${(\frac{3 (0 + 0)}{3 (0 + 0 \cdot 4 + 1)})}^{3}$

Step 11.2.1.5.7

Rewrite the expression.

${(\frac{0 + 0}{0 + 0 \cdot 4 + 1})}^{3}$

Step 11.2.2

Add $0$ and $0$ .

${(\frac{0}{0 + 0 \cdot 4 + 1})}^{3}$

Step 11.2.3

Simplify the denominator.

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Step 11.2.3.1

Multiply $0$ by $4$ .

${(\frac{0}{0 + 0 + 1})}^{3}$

Step 11.2.3.2

Add $0$ and $0$ .

${(\frac{0}{0 + 1})}^{3}$

Step 11.2.3.3

Add $0$ and $1$ .

${(\frac{0}{1})}^{3}$

Step 11.2.4

Divide $0$ by $1$ .

$0^{3}$

Step 11.2.5

Raising $0$ to any positive power yields $0$ .

$0$