if f' is continuous, f(2)=0 and f'(2)=7. evaluate the limit as x approaches 0 of f(2+3x)+f(2+5x)/x

As x approaches 0, 2 + 3x approaches 2 + 0= 2 and lim f(2 + 3x) as x approaches 0 = f (2)= 0

Similarly lim f(2 + 5x) as x approaches 0 = f (2)= 0

So since lim f(2+5x)/x = 0/0 and f and x are both differentiable,

lim f(2+5x)/x = lim f'(2 +5x)/1 as x approaches 0 = f'(2)=7

Hence the limit as x approaches 0 of f(2+3x)+f(2+5x)/x = 0 + 7= 7

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