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

Gatineau

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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'd start by using formula f(x+h)=f(x)+f(h)+O(h2)f(x + h) = f(x) + f^\prime(h) + O(h^2).
 
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:

limh0[f(5+6h2)f(52h2)][g(5+h)g(5h)]h3\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}

This might be split apart as:

g(5+h)g(5h)h\displaystyle \frac{g(5+h)-g(5-h)}{h}

...and:

f(5+6h2)f(52h2)h2\displaystyle \frac{f(5+6h^2) - f(5-2h^2)}{h^2}

Eliz.
 
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