Question

How do you determine the limit of `(x+2)/(x+3)` as x approaches `-3^+`?

Answer 1

`-oo`

Explanation:

`lim_{x \to -3^+}(x+2)/(x+3)`

if we just plug in `x = -3`, we can see that it is `2/oo`. the graph shows that `lim_{x \to -3^+}(x+2)/(x+3) = - oo`

to see this, let `x = -3 + \epsilon` {ie just to right of x = -3], with `0 < \epsilon < < 1` we have

`(-3 + \epsilon + 2)/(-3 + \epsilon + 3)`
`= ( -1 + \epsilon )/(\epsilon)`
`= ( -1)/(\epsilon) + (\epsilon )/(\epsilon)`
`= -( 1)/(\epsilon) + 1`

we see that the dominant term for small `\epsilon` is `= -( 1)/(\epsilon)`, which is negative; and `lim_{\epsilon \to 0} -( 1)/(\epsilon) + 1 = -oo`