how do I graph lim(x^2-4/x-2)

Question

I need to figure out how to graph this and determine the limit.

Answer 1

x²-4=(x-2)(x+2) so (x²-4)/(x-2)=(x-2)(x+2)/(x-2)=x+2 except when x=2, because that would produce 0/0 which is undefinable.

So the graph is just the same as y=x+2 (a straight line passing through points (0,2) and (-2,0)). Draw this line.

When drawing this graph as y=(x²-4)/(x-2) you need to mark the point (2,4) as a "hole" (draw a tiny circle on the line at this point--when x=2, y=4 because y=x+2=2+2=4). The limit is x+2 because for all points other than (2,4) the line exists. At (2,4) the line doesn't exist. A point has no dimensions but near to this point (as close as possible) the line does exist. Sometimes the hole is referred to as a singularity.

You can prove this by using your calculator: if you try to evaluate (x²-4)/(x-2) when x=2, your calculator will report an error; but if you evaluate the expression with, say, x=1.999, your calculator will give you the result 3.999.