Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.1.1

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

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x^{3}$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

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

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

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

Step 2.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 4

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

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

Step 5

Evaluate the limit.

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Step 5.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

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

Step 5.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

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

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

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

Step 7

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

$\frac{2 \cdot 0 - 0}{1 + 0 + 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $2$ by $0$ .

$\frac{0 - 0}{1 + 0 + 0}$

Step 8.1.2

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{1 + 0 + 0}$

Step 8.1.3

Add $0$ and $0$ .

$\frac{0}{1 + 0 + 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Add $1 + 0$ and $0$ .

$\frac{0}{1 + 0}$

Step 8.2.2

Add $1$ and $0$ .

$\frac{0}{1}$

Step 8.3

Divide $0$ by $1$ .

$0$