What are the asymptotes and removable discontinuities, if any, of f(x)= (x^2+4)/(x-3) f(x)=x2+4x−3?

1 Answer
Topscooter
Dec 25, 2015

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

Explanation:

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

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