The lim x -> 3 of (x^2 + 2x - 30)/sqrt(x - 3).
Here is what I would do:
Since you can't use direct substitution (would put a 0 in the denominator), first square both the top and bottom to get the square root out of the bottom. Then pick a value just to the left of the limit, such as 2.9, and a value just to the right of the limit, such as 3.1.
Your new equation after squaring both the top and bottom is:
(x^4 + 4x^3 - 56x^2 - 120x + 900)/(x - 3)
Now you can use the values to the left and right of three to plug in and see what if you get an overall negative or positive answer:
The lim x ->3 from the left side:
((2.9)^4 + 4(2.9)^3 - 56(2.9)^2 - 120(2.9) + 900) / (2.9 - 3) = (249.3241 / -0.1) =
(positive / negative) = negative infinity
The lim x -> 3 from the right side:
((3.1)^4 + 4(3.1)^3 - 56(3.1)^2 - 120(3.1) + 900) / (3.1 - 3) = (201.3561 / .1) =
(positive / positive) = positive infinity
Since the lim x -> 3 from the left side does not equal the lim x -> 3 from the right side, the limit does not exist.
Hope this helped, there may be an easier way to do this, but ive never seen a limit problem like this one with only a sqrt in the denominator. This answer should be right though.