Question

Does the limit as x → -2 of f(x) = (2x+4)/(x+2) exist?

Well, for it to exist the function must approach the same value from both sides, right? So if I do something crude like entering small intervals to both sides of -2 into the function (like -2.01 and -1.99) then I get values of -404.02 and 396.02, respectively. So this probably means the limit doesn't exist, right?

But wait, the function has a vertical asymptote at x=-2; doesn't this immediately mean that there's no limit as the function goes to - infinity on the one side and + infinity on the other? Then again, this might... (cont.)

Answer 1

... not be the case as it might also go to + or - infinity on both sides... But it doesn't in this case, so... hmm.

Also, let's say I can't use a calculator to just enter values -- what would I do to determine whether this limit exists?

Sorry for the essay. :P

Answer 2

yes it does

Answer 3

The function approaches the same value on both sides.

Answer 4

it is 2 since
\[\frac{2x+4}{x+2}=\frac{2(x+2)}{x+2}=2\]

Answer 5

... I'm an idiot >___>

Answer 6

well to be more precise,
\(\frac{2x+4}{x+2}=\frac{2(x+2)}{x+2}=2\) if \(x\neq -2\) but since this is a limit, we are asking what this gets close to when x is near but no equal to -2, so the limit is 2 for sure

Answer 7

http://www.wolframalpha.com/input/?i=%282%28-2.0001%29%2B4%29%2F%28%28-2.0001%29%2B2%29+%2C++%282%28-1.9999%29%2B4%29%2F%28%28-1.9999%29%2B2%29

Answer 8

Oh, I see... how did I even get those values? Must have entered something wrong... sorry :[
Thanks for the replies.