How can the limit of a function be determined without using a graph?

[BayernBlues]

Homework Statement

http://img442.imageshack.us/img442/7875/pset33po8.png

Homework Equations

The Attempt at a Solution

I know that the answer is limit does not exist but I don't know how to show it mathematically on paper. Is there a way to show this without making a graph?

 

[turdferguson]

Plug in 2. You get a nonzero number over 0, which means dividing a common factor won't help. The graph is asymtotic at x=2. You can show which infinity the function goes to with a sign graph. Find all important numbers (roots, asymtotes) and evaluate the function in between each number. The graph will only change signs possibly at a root or asymtote. By plugging in values close to 2 on either side. You should see that the limit is split and does not exist

 

[Werg22]

A quick way to check this is to see how does the numerator behave with x -> 2+ and x -> 2-, i.e.

2^3 + 3*2^2 + 4 = 8. Obviously, if x - > 2+, x^3 + 3x^2 + 4 > 8. Now,

x^3 + 3x^2 + 4 > 8

x^3 + 3x^2 + 4 / x - 2 > 8 / x - 2

Since on the right hand side the limit is + infinity, by the inequality we conclude that the limit on left is also + infinity. Now you can proceed with 2-... it may seem a bit long on paper, but it easily done mentally.