Precalculus Examples

$lim_{x \to \infty} \frac{9 x^{4} + 3}{4 x^{4}}$

Step 1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to \infty} \frac{9 x^{4} + 3}{x^{4}}$

Step 2

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

$\frac{1}{4} lim_{x \to \infty} \frac{\frac{9 x^{4}}{x^{4}} + \frac{3}{x^{4}}}{\frac{x^{4}}{x^{4}}}$

Step 3

Evaluate the limit.

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Step 3.1

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

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Step 3.1.1

Cancel the common factor.

$\frac{1}{4} lim_{x \to \infty} \frac{\frac{9 x^{4}}{x^{4}} + \frac{3}{x^{4}}}{\frac{x^{4}}{x^{4}}}$

Step 3.1.2

Divide $9$ by $1$ .

$\frac{1}{4} lim_{x \to \infty} \frac{9 + \frac{3}{x^{4}}}{\frac{x^{4}}{x^{4}}}$

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

$\frac{1}{4} lim_{x \to \infty} \frac{9 + \frac{3}{x^{4}}}{\frac{x^{4}}{x^{4}}}$

Step 3.2.2

Rewrite the expression.

$\frac{1}{4} lim_{x \to \infty} \frac{9 + \frac{3}{x^{4}}}{1}$

Step 3.3

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

$\frac{1}{4} \cdot \frac{lim_{x \to \infty} 9 + \frac{3}{x^{4}}}{lim_{x \to \infty} 1}$

Step 3.4

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

$\frac{1}{4} \cdot \frac{lim_{x \to \infty} 9 + lim_{x \to \infty} \frac{3}{x^{4}}}{lim_{x \to \infty} 1}$

Step 3.5

Evaluate the limit of $9$ which is constant as $x$ approaches $\infty$ .

$\frac{1}{4} \cdot \frac{9 + lim_{x \to \infty} \frac{3}{x^{4}}}{lim_{x \to \infty} 1}$

Step 3.6

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

$\frac{1}{4} \cdot \frac{9 + 3 lim_{x \to \infty} \frac{1}{x^{4}}}{lim_{x \to \infty} 1}$

Step 4

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

$\frac{1}{4} \cdot \frac{9 + 3 \cdot 0}{lim_{x \to \infty} 1}$

Step 5

Evaluate the limit.

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Step 5.1

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

$\frac{1}{4} \cdot \frac{9 + 3 \cdot 0}{1}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Divide $9 + 3 \cdot 0$ by $1$ .

$\frac{1}{4} (9 + 3 \cdot 0)$

Step 5.2.2

Multiply $3$ by $0$ .

$\frac{1}{4} (9 + 0)$

Step 5.2.3

Add $9$ and $0$ .

$\frac{1}{4} \cdot 9$

Step 5.2.4

Combine $\frac{1}{4}$ and $9$ .

$\frac{9}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{4}$

Decimal Form:

$2.25$