Calculus Examples

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

Step 1

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.2.4

Divide $x^{2}$ by $1$ .

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

Step 3

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

$lim_{x \to \infty} \frac{\sqrt{x^{2} + 0}}{\frac{2}{x} - 3}$

Step 4

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

$lim_{x \to \infty} \frac{\sqrt{x^{2} + 0}}{0 - 3}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{\sqrt{x^{2} + 0}}{0 - 3}$ approaches negative infinity.

$- \infty$