Domain, range, vertical & horizontal asymptotes, and removable discontinuity of a rational function

How to find the domain, range, removable discontinuity, vertical asymptote, and horizontal asymptote of a rational function. This is a must-know topic for both of your precalculus and calculus 1 classes. Subscribe to @bprpcalculusbasics for more calculus basics.

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[carultch]

Add Oblique Asymptotes to this list as well, for when the rational function is an improper fraction, with a degree on the top, that is one more than the degree on the bottom.

[Ninja]

Ive always found it easier to write the domain and range using the takeaway notation rather than with unions if you only need to exclude singular numbers and not intervals.

So you could write
Domain: R\{2, 4}
Range: R\{-3/2, 0}

which would be a lot shorter. But thats just preference.

[danilojonic]

I see why this approach would be easy for newbies but it's not rigorous at all. For vertical asymptotes, you take a limit at those critical points of the domain but because x = 2 gives indeterminate form (0/0), we would have to use Lhopital's rule which is a more advanced technique I guess. I know my approach might require more knowledge but it's also much more convenient and easier to explain. For vertical asymptote at 2, we take a limit when x gradually approaches value of 2. and there we get a specific value of -3/2 for both left and right limit meaning the graph from both sides will come towards -3/2. On the other side, left and right limit at x = 4, give us infinity meaning it's an asymptote here since graph catapults into infinity.

[DavidSeungLee]

Lol, I just learned about rational functions and limits when x approaches a hole in my precalculus class