Calculus Examples

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

Step 1

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

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

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

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Divide $3$ by $1$ .

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

Step 3.1.2

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

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Step 3.1.2.1

Factor $x$ out of $2 x$ .

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

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Cancel the common factor.

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

Step 3.1.2.2.3

Rewrite the expression.

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

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

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Step 3.2.2.1

Factor $x$ out of $- 3 x$ .

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

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.3

Rewrite the expression.

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 4

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

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

Step 5

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

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

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 $\infty$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

${(\frac{3 + 2 \cdot 0 + 0}{1 - 3 \cdot 0 + 2 \cdot 0})}^{4}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $2$ by $0$ .

${(\frac{3 + 0 + 0}{1 - 3 \cdot 0 + 2 \cdot 0})}^{4}$

Step 10.1.2

Add $3 + 0$ and $0$ .

${(\frac{3 + 0}{1 - 3 \cdot 0 + 2 \cdot 0})}^{4}$

Step 10.1.3

Add $3$ and $0$ .

${(\frac{3}{1 - 3 \cdot 0 + 2 \cdot 0})}^{4}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $- 3$ by $0$ .

${(\frac{3}{1 + 0 + 2 \cdot 0})}^{4}$

Step 10.2.2

Multiply $2$ by $0$ .

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

Step 10.2.3

Add $1 + 0$ and $0$ .

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

Step 10.2.4

Add $1$ and $0$ .

${(\frac{3}{1})}^{4}$

Step 10.3

Divide $3$ by $1$ .

$3^{4}$

Step 10.4

Raise $3$ to the power of $4$ .

$81$