Calculus Examples

$lim_{x \to - \infty} \frac{7 x^{2} - 9}{4 x + 3}$

Step 1

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

$lim_{x \to - \infty} \frac{\frac{7 x^{2}}{x} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

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 $7 x^{2}$ .

$lim_{x \to - \infty} \frac{\frac{x (7 x)}{x} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

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 $1$ .

$lim_{x \to - \infty} \frac{\frac{x (7 x)}{x^{1}} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.1.1.2.2

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

$lim_{x \to - \infty} \frac{\frac{x (7 x)}{x \cdot 1} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.1.1.2.3

Cancel the common factor.

$lim_{x \to - \infty} \frac{\frac{x (7 x)}{x \cdot 1} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.1.1.2.4

Rewrite the expression.

$lim_{x \to - \infty} \frac{\frac{7 x}{1} + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.1.1.2.5

Divide $7 x$ by $1$ .

$lim_{x \to - \infty} \frac{7 x + \frac{- 9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.1.2

Move the negative in front of the fraction.

$lim_{x \to - \infty} \frac{7 x - \frac{9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$lim_{x \to - \infty} \frac{7 x - \frac{9}{x}}{\frac{4 x}{x} + \frac{3}{x}}$

Step 2.2.2

Divide $4$ by $1$ .

$lim_{x \to - \infty} \frac{7 x - \frac{9}{x}}{4 + \frac{3}{x}}$

Step 3

As $x$ approaches $- \infty$ , the fraction $\frac{9}{x}$ approaches $0$ .

$lim_{x \to - \infty} \frac{7 x - 0}{4 + \frac{3}{x}}$

Step 4

As $x$ approaches $- \infty$ , the fraction $\frac{3}{x}$ approaches $0$ .

$lim_{x \to - \infty} \frac{7 x - 0}{4 + 0}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{7 x - 0}{4 + 0}$ approaches negative infinity.

$- \infty$