passionate
New member
- Joined
- Nov 15, 2007
- Messages
- 17
Could someone help me with this problem?
Let f: (-1,1) to R, and f'(0) exists. If the sequences (a_n), (b_n) converges to 0 , then define the difference quotion D_n=( f(b_n) - f(a_n) ) / ( b_n - a_n)
1) Show that lim D_n = f'(0) as n approaches infinity under the following condition
a) a_n < 0 < b_n
b) 0 < a_n < b_n and b_n / ( b_n - a_n ) < or equal to M
c) f'(x) exists and is continuous for all x in (-11)
2) let f(x)= x^2 * sin(1/x) for x does not equal 0 and f(0)=0. f is differentiable everywhere in (-1,1) and f'(0)=0. Find a_n , b_n that tend to 0 in such a way that D_n converges to a limit unequal to f'(0)
Let f: (-1,1) to R, and f'(0) exists. If the sequences (a_n), (b_n) converges to 0 , then define the difference quotion D_n=( f(b_n) - f(a_n) ) / ( b_n - a_n)
1) Show that lim D_n = f'(0) as n approaches infinity under the following condition
a) a_n < 0 < b_n
b) 0 < a_n < b_n and b_n / ( b_n - a_n ) < or equal to M
c) f'(x) exists and is continuous for all x in (-11)
2) let f(x)= x^2 * sin(1/x) for x does not equal 0 and f(0)=0. f is differentiable everywhere in (-1,1) and f'(0)=0. Find a_n , b_n that tend to 0 in such a way that D_n converges to a limit unequal to f'(0)