Rough introduction to limits

Summary

• Given:
• a real-valued function of a real variable, \(f: A→B\)
• a real number \(x_0\) where \(x_0\) may or may not belong to \(A\)
• all real numbers “near” \(x_0\) are in \(A\)
• Example:

\[f\left( x \right)= \frac{x^2-1}{x-1}\]

• with its natural domain and \(x_0=1\) . \(x_0∉A\) but all real numbers “ near ” \(x_0=1\) are in \(A\) .
• We say that \(f(x)\) approaches the limit \(L\) as \(x \) approaches \(x_0\) if you can make \(f(x)\) arbitrarily close to \(L\) by selecting \(x\) close to \(x_0\) .
• Example: \(f\left( x \right)\) approaches \(L=2\) when \(x \) approaches \(1\)
• If \(f(x)\) approaches the limit \(L\) as \(x \) approaches \(x_0\) then we write

\[\lim_{x→x_0} f(x)=L\]

• or

\[f\left( x \right)→L as x→x_0\]