Find the Asymptotes 1/(e^x-1)
The task is to determine the lines called asymptotes for the function f(x) = 1/(e^x - 1), where e represents the base of the natural logarithm. Asymptotes are straight lines that a curve approaches as it heads towards infinity or negative infinity. The question essentially asks to identify any horizontal, vertical, or oblique lines that the graph of the given function approaches, but does not actually touch, as the independent variable x either increases without bound (x -> ∞) or decreases without bound (x -> -∞).
$\frac{1}{e^{x} - 1}$
Answer
Solution:
Step 1:
Identify points where $\frac{1}{e^{x} - 1}$ is not defined. This occurs when $x = 0$.
Step 2:
As the denominator grows without bound, the value of $\frac{1}{e^{x} - 1}$ tends towards $0$.
Step 3:
Determine the horizontal asymptote by calculating $\lim_{x \to -\infty} \frac{1}{e^{x} - 1}$.
Step 3.1:
Proceed to evaluate the limit.
Step 3.1.1:
Apply the Limit Quotient Rule as $x$ approaches negative infinity: $\frac{\lim_{x \to -\infty} 1}{\lim_{x \to -\infty} e^{x} - 1}$.
Step 3.1.2:
The limit of the constant $1$ as $x$ approaches negative infinity is: $\frac{1}{\lim_{x \to -\infty} e^{x} - 1}$.
Step 3.1.3:
Utilize the Sum of Limits Rule as $x$ approaches negative infinity: $\frac{1}{\lim_{x \to -\infty} e^{x} - \lim_{x \to -\infty} 1}$.
Step 3.2:
As $x$ goes to negative infinity, $e^{x}$ tends to $0$: $\frac{1}{0 - \lim_{x \to -\infty} 1}$.
Step 3.3:
Finalize the evaluation of the limit.
Step 3.3.1:
The limit of the constant $1$ as $x$ approaches negative infinity is: $\frac{1}{0 - 1}$.
Step 3.3.2:
Simplify the expression.
Step 3.3.2.1:
Work on simplifying the denominator.
Step 3.3.2.1.1:
Multiply $-1$ by $1$: $\frac{1}{0 - 1}$.
Step 3.3.2.1.2:
Subtract $1$ from $0$: $\frac{1}{-1}$.
Step 3.3.2.2:
Divide $1$ by $-1$ to get $-1$.
Step 4:
Enumerate the horizontal asymptotes: $y = 0$ and $y = -1$.
Step 5:
An oblique asymptote is not present as the numerator's degree is not greater than the denominator's degree.
Step 6:
Compile the complete list of asymptotes:
Vertical Asymptotes: $x = 0$ Horizontal Asymptotes: $y = 0$ and $y = -1$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a function, we must consider vertical, horizontal, and oblique asymptotes. The process involves the following concepts:
Vertical Asymptotes: These occur where the function is undefined, typically where the denominator of a fraction is zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by evaluating the limit of the function as $x$ approaches positive or negative infinity. If the limit exists, that value is the horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is one more than the degree of the denominator. To find an oblique asymptote, divide the numerator by the denominator using polynomial division.
Limits and Continuity: Understanding limits is crucial for finding horizontal asymptotes. The Limit Quotient Rule states that if $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ both exist, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$. The Sum of Limits Rule states that $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.
Exponential Functions: The function $e^{x}$ approaches zero as $x$ approaches negative infinity, which is used to determine horizontal asymptotes for functions involving exponentials.
Asymptote Identification: After finding potential vertical and horizontal asymptotes, it's important to verify them by checking the behavior of the function around those points. Oblique asymptotes are checked by comparing the degrees of the numerator and denominator.
