Rough introduction to infinite limits

Summary

• Suppose that \(f\) is defined for all \(x\) "near" \(x_0\) (but not necessarily at \(x_0\) ). We then say that \(f(x)\) has the limit \(∞\) as \(x\) approaches \(x_0\) and write

\(\lim_{x→x_0}f(x)=∞\)

• if \(f(x)\) can be made arbitrarily large as \(x\) tends to \(x_0\) .
• \(f(x)→-∞\) as \(x→x_0\) is defined similarly.
• \(f(x)→±∞\) as \(x→±∞\) is defined similarly.
• Examples:
• \( \frac{1}{x^2}→∞\) as \(x→0\)
• \(logx→-∞\) as \(x→0+\)
• \( \frac{1}{x}→∞\) as \(x→0+\) while \( \frac{1}{x}→-∞\) as \(x→0-\)
• \(e^x→∞\) as \(x→∞\)
• Formal definition:
• Suppose that there is an open interval \((a,b)\) containing \(x_0\) such that \(f\) is defined for all \(x\) in this interval possibly with the exception of \(x_0\) . Then \(f(x)\) approaches the limit \(∞\) as \(x\) approaches \(x_0\) if for every \(c\) , there exists a \(δ>0\) such that \(f(x)>c\) whenever \(x_0-δ<x<x_0+δ\) .