# How to solve #lim_(x->a)(f(x)g(a)-f(a)g(x))/(x-a)#, if #f#,#g# have derivative on #x=a# ?

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To solve the limit lim_(x->a)(f(x)g(a)-f(a)g(x))/(x-a) when f and g have derivatives at x=a, we can use the limit definition of the derivative.

Let's denote f'(a) as the derivative of f at x=a, and g'(a) as the derivative of g at x=a.

Using the limit definition of the derivative, we have:

f'(a) = lim_(x->a) (f(x) - f(a))/(x-a) g'(a) = lim_(x->a) (g(x) - g(a))/(x-a)

Now, let's simplify the given limit expression:

lim_(x->a) (f(x)g(a) - f(a)g(x))/(x-a)

= lim_(x->a) (f(x)g(a) - f(a)g(a) + f(a)g(a) - f(a)g(x))/(x-a)

= lim_(x->a) (g(a)(f(x) - f(a))/(x-a) + f(a)(g(a) - g(x))/(x-a))

= g(a) * lim_(x->a) (f(x) - f(a))/(x-a) + f(a) * lim_(x->a) (g(a) - g(x))/(x-a)

= g(a) * f'(a) + f(a) * g'(a)

Therefore, the solution to the given limit is g(a) * f'(a) + f(a) * g'(a).

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