Math / Calculus

QuestionFind all horizontal asymptotes of the function $f(x) = \frac{1-e^{x}}{1+2 e^{x}}$ .

Studdy Solution

STEP 1

1. A horizontal asymptote is a horizontal line that the graph of a function approaches as $x \to \infty$ or $x \to -\infty$ .
2. To find horizontal asymptotes, we need to evaluate the limits of $f(x)$ as $x \to \infty$ and $x \to -\infty$ .

STEP 2

1. Evaluate the limit of $f(x)$ as $x \to \infty$ .
2. Evaluate the limit of $f(x)$ as $x \to -\infty$ .
3. Identify the horizontal asymptotes based on these limits.

STEP 3

Evaluate the limit of $f(x) = \frac{1-e^{x}}{1+2 e^{x}}$ as $x \to \infty$ .
As $x \to \infty$ , $e^x$ becomes very large. Therefore, both the numerator and the denominator are dominated by the terms involving $e^x$ .
$\lim_{x \to \infty} \frac{1-e^{x}}{1+2 e^{x}} = \lim_{x \to \infty} \frac{-e^{x}}{2e^{x}} = \lim_{x \to \infty} \frac{-1}{2} = -\frac{1}{2}$

STEP 4

Evaluate the limit of $f(x) = \frac{1-e^{x}}{1+2 e^{x}}$ as $x \to -\infty$ .
As $x \to -\infty$ , $e^x$ approaches 0. Therefore, the function simplifies to:
$\lim_{x \to -\infty} \frac{1-e^{x}}{1+2 e^{x}} = \frac{1-0}{1+0} = 1$

STEP 5

Based on the limits calculated, the horizontal asymptotes of the function are:
As $x \to \infty$ , the horizontal asymptote is $y = -\frac{1}{2}$ .
As $x \to -\infty$ , the horizontal asymptote is $y = 1$ .
The horizontal asymptotes of the function are $y = -\frac{1}{2}$ and $y = 1$ .

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