
Limit on Mars.
I've managed to find a solution to the following question using the f(a+h)  f(a) / h definition of the tangent line limit, but I got caught up trying to get the same answer using the: lim x > 1 f(x)  f(1) / x1 definition:
If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t  1.86t^2
(a) find the velocity of the rock after one second (I got 6.28 m/s)
(b) find the velocity of the rock when t = a (I got 103.72a)
The question doesn't ask about the second definition but I'm just curious to know what mistake I made while trying to calculate it using the f(x)  f(1) definition. Thanks for any clarification :)

might be easier to identify a mistake if you show your attempt ...

Hey skeeter, thanks for the reply! I was hesitant to show my work (some people can be harsh on forums). From what I can read of my scribbleheavy writing, this was the last step I got:
lim x > 1 (10x  1.86x^2  (10  1.86))/ x1
then
lim x > 1 (1.86x^2 + 10x  8.14 )/ x1
I tried to factor out a 1.86 hoping to be able to work with what was in the brackets to get an "x1" term, but this is the last thing I have in my notes...
lim x > 1 1.86(x^2  (10/1.86)x + (8.14/1.86)) / x1

Basic rule of algebra: If, for polynomial p(x), p(a)= 0, then x a is a factor of p(x).
Note that when x= 1, f(x) f(1)= f(1) f(1)= 0.
Specifically, when x= 1, 10(1) 1.86(1^2) (10 1.86)= 10 1.86 10+ 1.86= 0
so that 186(1^2)+ 10(1) 8.14= 0 which tells you that x 1 is a factor of 1.86x^2+ 10x 8.14! Divide that quadratic by x 1 to find the other factor, P(x). Once you know that, your limit becomes $\displaystyle \lim_{x\to 1} \frac{(x1)P(x)}{x1}= \lim_{x\to 1}P(x)$.