Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Factor $x$ out of $- x$ .

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

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 2

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

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

Step 3

Simplify each term.

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

Step 4

Cancel the common factors.

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{2 x + 3}{\sqrt{\frac{x - 1}{x}}}$ approaches infinity.

$\infty$