
Equation of tangent
Hello im stuck with a problem and the ONLY way i wanna solve it with a formel we suposed to use not derivate.
$\displaystyle Y=\frac{x1}{x2}$ on point (3,2)
the formel ima use is lim x>0 $\displaystyle \frac {f(a+h)f(a)}{h}$and a=3
Progress:
$\displaystyle \frac{a+h1}{a+h1}2/h$ it is suposed to mean that all that divide by h im trying my best with latex does not work well (notice i rewrite (31)/(32) as 2...
=$\displaystyle \frac{a+h12a2h+4}{a+h2}/h$ (notice that h is suposed to divide all that i mean like (a/b)/(c/d) and h is the c and that in latex code is a at top and b the one down.
this is where i notice this dont work over and over... i am doing something wrong that i cant see

Re: Equation of tangent
I've been out of the game for a long time so you can check my algebra to see that it's correct.
If $\displaystyle f(x)=\frac{x1}{x2}$, then
$\displaystyle f(x+h)f(x) = \frac{x+h1}{x+h2}  \frac{x1}{x2} = \frac{(x+h1)(x2)  (x1)(x+h2)}{(x+h2)(x2)} = \frac{x^2+xhx2x2h+2(x^2+xh2xxh+2)}{(x+h2)(x2)} = \frac{h}{(x+h2)(x2)}$
hence
$\displaystyle \frac{f(x+h)f(x)}{h} = \frac{1}{(x+h2)(x2)}$
now take the limit as h goes to zero to find the equation for the slope of a tangent curve where the function f(x) is continuous ie. wherever x is not equal to 2. so, plug in x=3 after evaluating the limit
Of course, this is by using definition alone, but it's way easier if you prove the general case of curves of f(x) = g(x)/h(x) (which would give you quotient rule, which is a special case of product rule. essentially reprove product rule if you haven't in class can save some time) and then use substitution of g(x) = x1 and h(x) = x2.