Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $4$ by $0$ .

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

Step 6.1.2

Multiply $- 1$ by $1$ .

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

Step 6.1.3

Subtract $1$ from $0$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 1$ by $0$ .

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

Step 6.2.2

Add $1$ and $0$ .

$\frac{- 1}{1}$

Step 6.3

Divide $- 1$ by $1$ .

$- 1$