Question

True or False If \(f(x)\) has a vertical asymptote at \(x=c\) , then either \(\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c^{+}} f(x)=\infty\) or \(\lim _{x \rightarrow c^{-}} f(x)=\) \(\lim _{x \rightarrow c^{+}} f(x)=-\infty .\) Justify your answer.

Short answer

The statement is False. The limits of \(f(x)\) as \(x\) approaches \(c\) from the left and right do not necessarily have to be the same, positive or negative infinity, for a vertical asymptote to exist at \(x=c\).

Step-by-step solution

Explanation of Vertical Asymptotes

A vertical asymptote of a function is a vertical line \(x=c\) where the values of \(f(x)\) approach infinity or negative infinity as \(x\) approaches \(c\) either from the right or the left.

Analysis of the Statement

The statement suggests that if \(f(x)\) has a vertical asymptote at \(x=c\), then the limit of \(f(x)\) as \(x\) approaches \(c\) from either direction is always positive or negative infinity. However, this is not necessarily always the case. Consider a function where \(f(x)\) approaches positive infinity as \(x\) approaches \(c\) from the right and \(f(x)\) approaches negative infinity as \(x\) approaches \(c\) from the left. This function would still have a vertical asymptote at \(x=c\), but the limits from the left and right are not the same.

Conclusion

Thus, the original statement is False. While a function having a vertical asymptote at \(x=c\) does imply that \(f(x)\) approaches either positive or negative infinity as \(x\) approaches \(c\), these limits do not need to be the same from the left and right.