Now given that, what would be the limit of f of x plus g of x as x approaches c? This is often called the sum rule, or the sum property, of limits.
Well-- and you could look at this visually, if you look at the graphs of two arbitrary functions, you would essentially just add those two functions-- it'll be pretty clear that this is going to be equal to-- and once again, I'm not doing a rigorous proof, I'm just really giving you the properties here-- this is going to be the limit of f of x as x approaches c, plus the limit of g of x as x approaches c. And we could come up with a very similar one with differences.
The “lim” shows limit, and fact that function f(n) approaches the limit L as n approaches c is described by the right arrow as: f(n) = L.
We assume that To consider the limit of a sum of difference, select the limits individually and put them back with the corresponding sign.
So let's say we know that the limit of some function f of x, as x approaches c, is equal to capital L.
And let's say that we also know that the limit of some other function, let's say g of x, as x approaches c, is equal to capital M.
So if we just had the limit-- let me do it in that same color-- the limit of k times f of x, as x approaches c, where k is just some constant. This is the exact same thing as the limit of f of x as x approaches c, divided by the limit of g of x as x approaches c.
This is going to be the same thing as k times the limit of f of x as x approaches c. Which is going to be equal to-- I think you get it now-- this is going to be equal to L over M.
And what's neat about it is the property of limits kind of are the things that you would naturally want to do.
And if you graph some of these functions, it actually turns out to be quite intuitive.