Calculus Examples

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

Step 1

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

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

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 $x^{3}$ and $x$ .

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Step 2.1.1.1

Factor $x$ out of $3 x^{3}$ .

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

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 $1$ .

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Cancel the common factor.

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

Step 2.1.1.2.4

Rewrite the expression.

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

Step 2.1.1.2.5

Divide $3 x^{2}$ by $1$ .

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

Step 2.1.2

Cancel the common factor of $x$ .

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Step 2.1.2.1

Cancel the common factor.

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

Step 2.1.2.2

Rewrite the expression.

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

Step 2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

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

$lim_{x \to \infty} \frac{3 x^{2} - 1 + 0}{1 + \frac{3}{x}}$

Step 4

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

$lim_{x \to \infty} \frac{3 x^{2} - 1 + 0}{1 + 0}$

Step 5

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{3 x^{2} - 1 + 0}{1 + 0}$ approaches infinity.

$\infty$