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Math Help - differentibility

  1. #1
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    differentibility

    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
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  2. #2
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    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?
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    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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