Calculus Examples

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

Step 1

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Cancel the common factor.

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

Step 2.1.2.4

Rewrite the expression.

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

Step 2.1.2.5

Divide $x$ by $1$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $2$ by $1$ .

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

Step 3

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

$lim_{x \to \infty} \frac{x + 0}{2 - \frac{1}{x}}$

Step 4

As $x$ approaches $\infty$ , the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 5

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

$\infty$