Question

Writing Consider the function \(f(x)=\frac{2}{1+e^{1 / x}}\) (a) Use a graphing utility to graph \(f\). (b) Write a short paragraph explaining why the graph has a horizontal asymptote at \(y=1\) and why the function has a nonremovable discontinuity at \(x=0\).

Short answer

The function \(f(x)=\frac{2}{1+e^{1/x}}\) has a horizontal asymptote at \(y=2\) rather than at \(y=1\) as \(x\) approaches positive or negative infinity. The function exhibits a nonremovable discontinuity at \(x=0\) because the function is undefined at this point, creating a hole in the graph.

Step-by-step solution

Graphing the function

To graph the function \(f(x)=\frac{2}{1+e^{1/x}}\), use a graphing calculator or online graphing tool. Alternatively, one can sketch the graph by plotting some key points and noting the behavior of the function as \(x\to 0^+\), \(x\to 0^-\), \(x\to +\infty\), and \(x\to -\infty\).

Identifying the horizontal asymptote

The graph has a horizontal asymptote at \(y=1\) because the function tends to \(1\) as \(x\) tends to positive or negative infinity. Mathematically, this is because the term \(e^{1/x}\) tends to \(0\) as \(x\) tends to positive or negative infinity, thus making the entire function to tend to \(2/(1+0) = 2/1 = 2\). But since the entire expression could only reach a maximum of 2, the line \(y=2\) becomes a ceiling that the function cannot cross, forming a horizontal asymptote at \(y=2\).

Explaining the nonremovable discontinuity

The function has a nonremovable discontinuity at \(x=0\) because the term \(e^{1/x}\) is undefined at \(x=0\). This means that the function \(f(x)\) does not exist at \(x=0\), creating a gap or a hole in the graph at this point, a characteristic of nonremovable discontinuity.