What are the asymptotes and removable discontinuities, if any, of #f(x)= (x^2+4)/(x-3) #?

1 Answer
Dec 25, 2015

No removable discontinuities, and the 2 asymptotes of this function are #x = 3# and #y = x#.

Explanation:

This function is not defined at #x = 3#, but you can still evaluate the limits on the left and on the right of #x = 3#.

#lim_(x->3^-)f(x) = -oo# because the denominator will be strictly negative, and #lim_(x->3^+)f(x) = +oo# because the denomiator will be strictly positive, making #x = 3# an asymptote of #f#.

For the 2nd one, you need to evaluate #f# near the infinities. There is a property of rational functions telling you that only the biggest powers matter at the infinities, so it means that #f# will be equivalent to #x^2/x = x# at the infinites, making #y = x# another asymptote of #f#.

You can't remove this discontinuity, the 2 limits at #x=3# are different.

Here is a graph :
graph{(x^2 + 4)/(x - 3) [-163.5, 174.4, -72.7, 96.2]}