When the limit isn't provided, how do I know when to use $\frac{f(x)-f(a)}{x-a}$ over $\frac{f(a+h)-f(a)}{h}$?

Question

I am unsure of which to pick when trying to solve a problem. For example; I was given the problem:

Find an equation of the tangent line to the curve at the given point

$$y=4x-3x^2, \qquad (2,-4)$$

which was then solved by the $(f(a+h)-f(a))/h$ formula.

I was then given the problem;

Find an equation of the tangent line to the curve at the given point;
$$y = \frac{2x+1}{x+2}, \qquad (1,1)$$

which was solved using the $(f(x)-f(a))(x-a)$.

So, how does one pick which formula to use to get the correct answer?

Answer 1

The two definitions are equivalent, so either will give you the correct answer. I don't think there's a hard rule about when to use either definition.

If you try one definition and it seems confusing, then try the other definition.