Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Expand $(2 - x) (2 + x)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $2$ by $2$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 + 2 x - x \cdot 2 - x \cdot x}$

Step 1.2.1.2

Multiply $2$ by $- 1$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 + 2 x - 2 x - x \cdot x}$

Step 1.2.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.3.1

Move $x$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 + 2 x - 2 x - (x \cdot x)}$

Step 1.2.1.3.2

Multiply $x$ by $x$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 + 2 x - 2 x - x^{2}}$

Step 1.2.2

Subtract $2 x$ from $2 x$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 + 0 - x^{2}}$

Step 1.2.3

Add $4$ and $0$ .

$lim_{x \to \infty} \frac{2 x^{2} + 1}{4 - x^{2}}$

Step 2

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

$lim_{x \to \infty} \frac{\frac{2 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - \frac{x^{2}}{x^{2}}}$

Step 3

Evaluate the limit.

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Step 3.1

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

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Step 3.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{2 x^{2}}{x^{2}} + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - \frac{x^{2}}{x^{2}}}$

Step 3.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{2 + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - \frac{x^{2}}{x^{2}}}$

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - \frac{x^{2}}{x^{2}}}$

Step 3.2.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{2 + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - 1 \cdot 1}$

Step 3.2.2

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{2 + \frac{1}{x^{2}}}{\frac{4}{x^{2}} - 1}$

Step 3.3

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

$\frac{lim_{x \to \infty} 2 + \frac{1}{x^{2}}}{lim_{x \to \infty} \frac{4}{x^{2}} - 1}$

Step 3.4

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

$\frac{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} \frac{4}{x^{2}} - 1}$

Step 3.5

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{2 + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} \frac{4}{x^{2}} - 1}$

Step 4

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

$\frac{2 + 0}{lim_{x \to \infty} \frac{4}{x^{2}} - 1}$

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 $\infty$ .

$\frac{2 + 0}{lim_{x \to \infty} \frac{4}{x^{2}} - lim_{x \to \infty} 1}$

Step 5.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 + 0}{4 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} 1}$

Step 6

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

$\frac{2 + 0}{4 \cdot 0 - lim_{x \to \infty} 1}$

Step 7

Evaluate the limit.

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Step 7.1

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

$\frac{2 + 0}{4 \cdot 0 - 1 \cdot 1}$

Step 7.2

Simplify the answer.

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Step 7.2.1

Add $2$ and $0$ .

$\frac{2}{4 \cdot 0 - 1 \cdot 1}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Multiply $4$ by $0$ .

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

Step 7.2.2.2

Multiply $- 1$ by $1$ .

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

Step 7.2.2.3

Subtract $1$ from $0$ .

$\frac{2}{- 1}$

Step 7.2.3

Divide $2$ by $- 1$ .

$- 2$