Slant asymptotes (proving it?)

Sep 2017
200
2
nj
Hey, so I know how to find whether there is a slant asymptotes by doing long division of the numberators degree > denominator, but why does the limit of it as x goes to infinity and negative infity of f (x) -mx-b =0.

That's the only thing I don't get. What and how do it tell us about slant asymptotes?
 
Sep 2017
200
2
nj
Oh. So take the difference in f (x) and the slant line gives us the limit ad f (x) approaches the slant which should be 0..
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
A "slant asymptote" is line such that the graph gets closer and closer to the graph as x goes to positive or negative infinity. If the graph is given by y= f(x) and the line is y= mx+ b. For any x, the vertical distance between the graph and the line is f(x)- (mx+ b)= f(x)- mx- b. That must go to 0 as x goes to positive or negative infinity.
 
Sep 2017
200
2
nj
Thank you! I was wondering what taking the different of f (x) and line of slant asymptotes did, and it makes sense now! Thanks a bunch.