differentibility

**henri** · November 3rd 2008, 09:07 AM

If f(x) is differentiable at x=a (a is not equal to 0), evaluate

lim (f^4 (x) - F^4 (x))/ x^4 - a^4 as x goes to a

**Plato** · November 3rd 2008, 09:38 AM

I think that you mean $\lim _{x \to a} \frac{{f^4 (x) - {\color{red}f}^4 ({\color{red}a})}}{{x^4 - a^4 }}$
Is that correct?

**chiph588@** · November 3rd 2008, 02:55 PM

Assuming Plato is right...

$\lim_{x \to a} \frac{f^{4}(x)-f^{4}(a)}{x^{4}-a^{4}} = \lim_{x \to a} \frac{(f(x)-f(a))(f(x)+f(a))(f^{2}(x)+f^{2}(a))}{(x-a)(x+a)(x^{2}+a^{2})}$ (by factoring)

$= \left(\lim_{x \to a} \frac{(f(x)-f(a))}{x-a}\right)\left(\lim_{x \to a} \frac{(f(x)+f(a))(f^{2}(x)+f^{2}(a))}{(x+a)(x^{2}+ a^{2})}\right)$

$= f'(a)\frac{(2f(a))(2f^{2}(a))}{(2a)(2a^{2})} = f'(a)\frac{f^{3}(a)}{a^{3}}$

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