Find the limit at x approaches 2?

Question

Find the limit at x approaches 2 of

(x^2 - 2) / (4 - x^2)

Please explain how you derive the answer. Thanks in advance.

Answer 1

Ok this a a tricky one.. The best way you can do this is to graph out your equation. Either use a calculator or a t-table.

When you graph it out you will have vertical asymptotes at: x=
-2,+2.. and a horizontal asymptote at y= -1. sooo.. do a number line analysis and you will find this:


from negitive infinity to -2.. it is negitive.. and from -2 to 2 it is positive and from 2 to positive infinity it is negitive.

and at the (2) mark.. the two points dont match at all. The dots aren't able to touch because its first off an asymptote and they are going in opposite directions. Therefore your limit Does not exisit (or is undefined) at 2 ( or -2 for that matter)

Answer 2

there is no thrick

putx=2+h,h+very small number.
then

quwestion is to find limit at h approaching 0

{(2+h)^2--2}/{4-(2+h)^2-}

={4+h^2-4h-2}/{ 4-4-h^2-4h}
h^2-4h+2/h(h-4)

which tends to -infinity
and if we take x=2-h it will tend to infinity(positive sense)

so function has a non removable discontinuity .Limit does not exist

Answer 3

substitute 2 in for x in the equation it is very simple
simplfies to 2/0 which is undefined. that limit is undefined.