Rough introduction to limits at infinity

Summary

• Suppose that \(f\) is defined for arbitrary large \(x\) -values. We then say that \(f(x)\) approaches the limit \(L\) as \(x\) tends to infinity and write

\(\lim_{x→∞}f(x)=L\)

• if \(f(x)\) can be made arbitrarily close to \(L\) by making \(x\) sufficiently large.

\(\lim_{x→-∞}f(x)=L\)

• is defined similarly
• Examples:
• \( \frac{1}{x}→0\) as \(x→∞\)
• \(e^x→0\) as \(x→-∞\)
• Formal definition:
• Suppose that \(f\) is defined for all \(x\) in \((a,∞)\) for some \(a\) . Then \(f(x)\) approaches the limit \(L\) as \(x\) approaches \(∞\) if for every number \(ε>0\) , there exists a \(c\) such that \(L-ε<f(x)<L+ε.\) whenever \(x>c\) .