t******g
发帖数: 1136 | 1 Can anyone give me any hint to following problem?
Thanks.
Suppose that $f(x)$ is continuous at $x_0$. And suppose $\{a_n\}$
and $\{b_n\}$ are any two sequences such that $a_n
$\lim_{n\rightarrow \infty}a_n=\lim_{n\rightarrow \infty}b_n=x_0$.
And
\[
\lim_{n\rightarrow \infty} \frac{f(b_n)-f(a_n)}{b_n-a_n}=L.
\]
Can we approve $f'(x_0)$ exists, and equal to $L$? Or find out a
counter example to show $f'(x_0)$ does not exist. | n***p
发帖数: 7668 | 2 No.
Counter example: f(x)=x sin(1/x) x_0 =0,
and a_n = -1/(2pi n), b_n = 1/(2pi n).
【在 t******g 的大作中提到】
: Can anyone give me any hint to following problem?
: Thanks.
: Suppose that $f(x)$ is continuous at $x_0$. And suppose $\{a_n\}$
: and $\{b_n\}$ are any two sequences such that $a_n: $\lim_{n\rightarrow \infty}a_n=\lim_{n\rightarrow \infty}b_n=x_0$.
: And
: \[
: \lim_{n\rightarrow \infty} \frac{f(b_n)-f(a_n)}{b_n-a_n}=L.
: \]
: Can we approve $f'(x_0)$ exists, and equal to $L$? Or find out a
| a****e
发帖数: 1247 | 3 晕， 那个题目里说的是any sequnces a_n , b_n.
【在 n***p 的大作中提到】
: No.
: Counter example: f(x)=x sin(1/x) x_0 =0,
: and a_n = -1/(2pi n), b_n = 1/(2pi n).
| n***p
发帖数: 7668 | 4 Ahh, then yes.
【在 a****e 的大作中提到】
: 晕， 那个题目里说的是any sequnces a_n , b_n.
|
|
