limites

Qual a pergunta??

Por hipótese, temos:

\(\exists M>0\ \vert0\leq\left\vert f(x)-f(p)\right\vert\leq M\left\vert x-p^2\right\vert\)

Dividindo por \(x-p\), temos:

\(0\leq{\left\vert f(x)-f(p)\right\vert\over x-p}\leq M{\left\vert x-p^2\right\vert\over x-p}\)

Calculando o limite, temos:

\(0\leq\lim\limits_{x\rightarrow p}{\left\vert f(x)-f(p)\right\vert\over x-p}\leq M\lim\limits_{x\rightarrow p}{\left\vert x-p^2\right\vert\over x-p}\)

Substituindo os valores, temos:

\(0\leq\lim\limits_{x\rightarrow p}{\left\vert f(x)-f(p)\right\vert\over x-p}\leq \infty\)

Temos, portanto, que o limite procurado diverge.