Formal def. of a lim. functional problem: [(f(5+6h^2)-f(5-2h^2)) * (g(5+h)-g(5-h))] / h^3

[Gatineau]

Given f'(5)=-2 and g'(5)=7
Evaluate the limit as h approaches zero of (f(5+6h^2 ) - f(5-2h^2 ))(g(5+h) - g(5-h))/h^3

I know this type of problem is just algebraic manipulation (like most limits), but I really didn't manage to solve it.

 

[blamocur]

I'd start by using formula [imath]f(x + h) = f(x) + f^\prime(h) + O(h^2)[/imath].

 

[stapel]

Given f'(5)=-2 and g'(5)=7
Evaluate the limit as h approaches zero of (f(5+6h^2 ) - f(5-2h^2 ))(g(5+h) - g(5-h))/h^3

I know this type of problem is just algebraic manipulation (like most limits), but I really didn't manage to solve it.

I *think* the limit is as follows:

[imath]\displaystyle \lim_{h \rightarrow 0} \frac{\left[f(5+6h^2) - f(5 - 2h^2)\right]\left[g(5+h) - g(5-h)\right]}{h^3}[/imath]

This might be split apart as:

[imath]\displaystyle \frac{g(5+h)-g(5-h)}{h}[/imath]

...and:

[imath]\displaystyle \frac{f(5+6h^2) - f(5-2h^2)}{h^2}[/imath]

Eliz.