Calculus Examples

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

Step 1

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

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

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 $x$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $0 - 1 \cdot 1$ and $0 + 1$ .

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Step 6.2.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{- 1 (0) - 1 \cdot 1}{0 + 1}$

Step 6.2.2

Factor $- 1$ out of $- 1 \cdot 1$ .

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

Step 6.2.3

Factor $- 1$ out of $- 1 (0) - 1 (1)$ .

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

Step 6.2.4

Cancel the common factor.

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

Step 6.2.5

Divide $- 1$ by $1$ .

$- 1$