Question

How do you find the limit of `(x-2)/(x^2+x-6)` x as it approaches pi/2?

Answer 1

Because `pi/2` does not make the denominator `0`, you can find the limit by substitution, but the arithmetic is simpler if we reduce first.

Explanation:

`(x-2)/(x^2+x-6) = (x-2)/((x+3)(x-2)) = 1/(x+3)`.

So,

`lim_(xrarrpi/2) (x-2)/(x^2+x-6) = lim_(xrarrpi/2)1/(x+3) = 1/(pi/2+3) = 2/(pi+6)`

Direct substitution will get the same answer, but making it look like this requires some algebra.

`(pi/2-2)/(pi^2/4+pi/2-6) = (2pi-8)/(pi^2+2pi-24)` `" "` (multiply by `4/4`)

`= (2(pi-4))/((pi+6)(pi-4))`

`= 2/(pi+6)`