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Limit

Limits

  • \(\epsilon\) method
    • sequence \(S_n\) goes to infinite \(L\) is limit of \(S_n\), iif there is \(N\) satisfing \(S_n\) \(-\) \(L < \epsilon\) in every \(n>N\).
  • Upper limit
    • \(\lim_{n\to\infty}\) \(sup(a_n)\)
  • Lower limit
    • \(-\lim_{n\to\infty}\) \(inf(a_n)\)

Limit of function

\(f_n\) \((x)\) \(=x^n\) (\(n \in N\))
\(f_n\) \(:(0,1) \rightarrow (0,1)\)

  • Pointwise limit

\(\lim_{n\to\infty}\) \(f_n\) \((x)\) \(=f(x) = 0\) for every \(x\)

for \(x=0.5\),
sequence of \(f_n\)\((0.5)\) is \(\{0.5, 0.25, 0.125 ...\}\)
\(\lim_{n\to\infty}\) \(f_n\) \((0.5)=0\)


  • Uniform limit

\(\sup_{n}\) \(\{\lvert f(x) - f_n(x) \rvert\ :x \in (0,1) \}\)
\(=\) \(\sup_{n}\) \(\{\lvert f_n(x) \rvert\ :x \in (0,1) \}\)

\(\sup_{1}\) \(\{\lvert x \rvert :x \in (0,1) \}\) \(=1\)
\(\sup_{2}\) \(\{\lvert x^2 \rvert :x \in (0,1) \}\) \(=1\)
\(\sup_{n\to\infty}\) \(\{\lvert x^n \rvert :x \in (0,1) \}\) \(=1\)

\(n>N\) such as \(\epsilon\) argument can not be determined because \(N\) changes according to \(x\) value.