# Hidden mistake

by limitkiller
Tags: hidden, mistake
 Sci Advisor HW Helper PF Gold P: 12,016 Hint: $$0=f(x+h)g(x)-f(x+h)(gx)$$
 PF Gold P: 864 Hidden mistake Just to expand on what other people have said, rules like "The limit of the difference is the difference of the limits" only apply when both limits exist. So it is not true that $$\lim_{h\to 0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}= \lim_{h\to 0}\frac{f(x+h)g(x+x)}{h} -\lim_{h\to 0}\frac{f(x)g(x)}{h} = \infty - \infty$$ (the last equality is assuming neither f nor g is 0 or has a 0 limit at x) Likewise, splitting up limits like that only works when the limits each exist for addition, multiplication and division. The limit of the denominator also can't be 0 in the case of division.