Finding the limits of this expression

[unscientific]

Homework Statement

I did two attempts. I find attempt no. 1 slightly fishy and attempt 2 more rigorous. Can anyone tell me what's wrong with attempt no. 1?

The Attempt at a Solution

Attempt no. 1:

Attempt no. 2:

 

[lurflurf]

They are both basically right. In one though
(F(x)-F(2))'=F'(x)-0=f(x)
F(2)' is not equal to F'(2)

This can be seen as the derivative of the anti-derivative of sin(x)/x

or the average of sin(x)/x over x near 2

in either case we get

sin(2)/2

 

[unscientific]

They are both basically right. In one though
(F(x)-F(2))'=F'(x)-0=f(x)
F(2)' is not equal to F'(2)

This can be seen as the derivative of the anti-derivative of sin(x)/x

or the average of sin(x)/x over x near 2

in either case we get

sin(2)/2

An example of F'(2) would be say...the first f' term in the taylor's expansion. F(2)' would be say, differentiating the constant term in taylor's expansion.

I see what you mean..thanks!