use power series method to obtain the specific solution to: u\'\'+cos(x)*u=0, u(

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I assume you are evaluating this limit as x approaches 0. We can use our knowledge of limits to rearrange what we know: lim [ (1-cosx)/ x ] = lim (1 - cos x) / lim (x) We know that lim (1 -cos x) = 0 as x -> 0 and lim ( x ) = 0 as x-> 0 This means that our entire limit would evaluate to 0/0 in the present form (aka indeterminate form). From this, we know that we can use L'Hospital's Rule in order to evaluate this limit. Using L'Hospital's Rule, we take the derivatives of both the numerator and the denominator seperately. Doing this, we get: lim ( sin(x) / 1) as x -> 0 We can easily see that this evaluates to 0 since sin(x) -> 0 as x -> 0.

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