Calculus Examples

$lim_{x \to \infty} \frac{9 x^{2} - 2}{6 - 29 x}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{9 x^{2}}{x} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2

Simplify terms.

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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 $x$ .

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Step 2.1.1.1

Factor $x$ out of $9 x^{2}$ .

$lim_{x \to \infty} \frac{\frac{x (9 x)}{x} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

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

$lim_{x \to \infty} \frac{\frac{x (9 x)}{x^{1}} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.1.2.2

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

$lim_{x \to \infty} \frac{\frac{x (9 x)}{x \cdot 1} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.1.2.3

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x (9 x)}{x \cdot 1} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.1.2.4

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{9 x}{1} + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.1.2.5

Divide $9 x$ by $1$ .

$lim_{x \to \infty} \frac{9 x + \frac{- 2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.1.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{9 x - \frac{2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{9 x - \frac{2}{x}}{\frac{6}{x} + \frac{- 29 x}{x}}$

Step 2.2.2

Divide $- 29$ by $1$ .

$lim_{x \to \infty} \frac{9 x - \frac{2}{x}}{\frac{6}{x} - 29}$

Step 3

As $x$ approaches $\infty$ , the fraction $\frac{2}{x}$ approaches $0$ .

$lim_{x \to \infty} \frac{9 x - 0}{\frac{6}{x} - 29}$

Step 4

As $x$ approaches $\infty$ , the fraction $\frac{6}{x}$ approaches $0$ .

$lim_{x \to \infty} \frac{9 x - 0}{0 - 29}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{9 x - 0}{0 - 29}$ approaches negative infinity.

$- \infty$