Question

Evaluate the following. If the limit does not exist, explain why. No marks will be given if l'Hospital's rule is used? `lim_(x->-2) (|x^2-4|)/(2x^2 + 3x -2)`

Answer 1

The limit does not exist.

Explanation:

`abs(x^2-4)/(2x^2 + 3x -2) = (abs(x+2)abs(x-2))/((x+2)(2x-1))`

For `x < 0`, we have

`abs(x+2)/(x+2) = {(1,"if",-2 < x < 0),(-1,"if",x < -2):}`

At `x = 2`, we see that `abs(x-2)/(2x-1) = abs(-4)/(-4-1) = -4/5`

Putting this together, we have

`abs(x^2-4)/(2x^2 + 3x -2) = {(-4/5,"if",-2 < x < 0),(4/5,"if",x < -2):}`

The left and right limits at `-2` are not equal.