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Question: a let and show that for all where...

Question details

(a) Let f EC(1) and x_{0}in I . Show that for all xin I:

f(x)=sum_{k=0}^{n}rac{f^{(k)}(x_{0})}{k!}(x-x_{0})^k+varphi (x)cdot (x-x_{0})^n,

where varphi is a function with limo)0 .

(b) Apply (a) to the Function f: (-1,1) ightarrow mathbb{R}, x mapsto sqrt{1+x} with; x_{0}=0, n=1 .

Prove with this that;

lim_{x ightarrow infty } (sqrt{x+sqrt{x}}-sqrt{x})= rac{1}{2}

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