Functie derivabila intr-un punct

[bogdanl_yex]

Fie \( f:R \rightarrow R \) o functie derivabila intr-un punct \( x_{0} \). Daca \( a_{n} \) este un sir strict crescator si convergent la \( x_{0} \) iar \( b_{n} \) este un sir strict descrescator si convergent la \( x_{0} \), atunci \( \lim_{n\rightarrow \infty}\frac{f(b_{n})-f(a_{n})}{b_{n}-a_{n}}=f^{\prime}(x_{0}) \).

[Bogdan Cebere]

\( \frac{f(b_{n})-f(a_{n})}{b_{n}-a_{n}}=\frac{f(b_{n})-f(x_0)}{b_n-x_0} \frac{b_{n}-x_0}{b_{n}-a_{n}}+\frac{f(x_{0})-f(a_n)}{x_0-a_n} \frac{x_0-a_n}{b_n-a_n} \). Cum \( \frac{f(b_{n})-f(x_0)}{b_n-x_0} \leq \frac{f(b_{n})-f(x_0)}{b_n-x_0} \frac{b_n-x_0}{b_n-a_n}+\frac{f(x_{0})-f(a_n)}{x_0-a_n} \frac{x_0-a_n}{b_n-a_n} \leq \frac{f(x_{0})-f(a_n)}{x_0-a_n} \) (deoarece \( \frac{b_n-x_0}{b_n-a_n}+\frac{x_0-a_n}{b_n-a_n} =1 \)) trecem la limita si obtinem
\( f^{\prime}(x_{0}) \leq \lim_{n\rightarrow \infty}\frac{f(b_{n})-f(a_{n})}{b_{n}-a_{n}} \leq f^{\prime}(x_{0}). \)