Multiply numerator and denominator by the conjugate:
I'm doing limit exercises and I got this question:
Evaluate the following limit:
x → −∞ √(x2 + 5 x + 1) - x
After working it out by multiplying by conjugate I get 5/0 which is undefined, but I know the answer is infinity. So where did I go wrong? This is the last step which I deduced 5/0 from:
(5 + (0))/-sqrt(1 + (0) + (0)) + 1
Any clarification is appreciated
Hey, thanks for the response! This was actually a 2 part exercise I was given for homework and I only posted the part (b). Part (a) was the exact same question but x goes to infinity rather than negative infinity. For that one I got 5/2 and the system told me I was right. But for the one I posted I got 5/0 but my graphing program showed it going to no particular horizontal asymptote so I had a sneaking suspicion that it was infinity. So that's what I submitted and it said I was correct :S ??
Also, in the denominator, since you're turning x into sqrt(x^2), aren't you losing a negative sign if you don't write it -sqrt(x^2)? Since x would be negative as x >> infinity?
Thank you Tonio, I did indeed get that but doesn't that end up being (5 + (0))/-sqrt(1 + (0) + (0)) + 1 which turns into 5/0? I've just started limits so I don't know if that is the correct way to finish the problem (obviously it's not since it gives wrong answer :P)
Well, as you know you can't divide by zero...
What you actually get is an expression of the form , with a constant and a
function which converges to zero. It's not hard to see, even by means of the definition, that the limit of
this expression has as limit , depending on the constant's and the function's signs.
Oh ok I will work on understanding that. We skipped the precise delta/epsilon description in our books and were just told to divide by greatest x degree in denominator. I just tried to go as far as I could with the definition. Maybe one day it will be as easy to see for me as it is for you guys! HAHA! Thanks!