Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Since the exponent $x$ approaches $- \infty$ , the quantity $e^{x}$ approaches $0$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Since the exponent $x$ approaches $- \infty$ , the quantity $e^{x}$ approaches $0$ .

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by $0$ .

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

Step 5.1.2

Add $1$ and $0$ .

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

Step 5.2

Add $1$ and $0$ .

$\frac{1}{1}$

Step 5.3

Divide $1$ by $1$ .

$1$