Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ f(x)=\frac{1}{e^{x}-1} $$

Short answer

The vertical asymptote of the function \( f(x)=\frac{1}{e^{x}-1} \) is \( x = 0 \).

Step-by-step solution

Write down the function

The function given is \( f(x)=\frac{1}{e^{x}-1} \).

Set the denominator equal to zero and solve for \( x \)

We need to find the vertical asymptote, that will occur when the denominator of this function is zero. So, we solve the equation \( e^{x} - 1 = 0 \). Solving for \( x \) we have \( x = \ln 1 \).

Simplify the solution

Since \( \ln 1 = 0 \), the vertical asymptote of the function is \( x = 0 \), because this is the value of \( x \) where the function tends to infinity or negative infinity.