Question

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0\)

Short answer

The statement 'If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0\)' is false.

Step-by-step solution

Understanding the concept of vertical asymptote

A vertical asymptote of a function is a vertical line \(x = a\) where the function \(f(x)\) approaches infinity or negative infinity as \(x\) approaches \(a\). It represents a value at which the function becomes infinitely large.

Deducing from the function

Despite the concept of vertical asymptote, it does not explicitly state that a function is undefined at the point of its vertical asymptote. For example, when we consider the function \(f(x) = (x^2 - 4)/(x - 2)\), rearranging the terms, we see that it is equivalent to \(f(x) = x + 2\), except for at \(x = 2\), it follows the line \(y = x + 2\) for all other values of \(x\) except \(x = 2\), so the function \(f(x)\) is undefined at \(x = 2\), even though there is a vertical asymptote at \(x = 2\).

Giving the Final Answer

Hence, it's possible to have a function with a vertical asymptote at a certain point without the function being undefined at that point. Therefore, the statement is false.