Delta Epsilon Proof Limitations?

[Taluric]

First off, I'm fresh into a Calc 1 class so my exposure to this is rather limited here and I am sorry if the answer is obvious.

The instructor today had us work a problem that was meant to be one sided as if it was two for no particularly good reason, the problem being:

F(x)= √(6+x)=0 as x approaches -6

Now, the proof is pretty simple... |√(6+x)-0|< (epsil) to |6+x|< (epsil)^2

But my problem is with the fact that we just proved it to exist with the 'precise definition' even though the limit does not exist from the left side without the use of imaginary numbers. I asked my instructor about that and he basically told me we should not have used that problem in that way, but he couldn't answer why we were able to prove a non-existent limit or give me any indication in the proof that would show this to be the case. Are we supposed to be doing approaches from both sides separately for 'real' proof or am I missing something else?

Thanks for reading.

 

[daon2]

Simply writing \(\displaystyle \sqrt{6+x}\) implicitly defines \(\displaystyle x\ge -6\) (As a function on the real numbers.)

So |x+6| = x+6.

As a complex function, it is still continuous at 0. But then the notion of "one sided limit" is thrown out the window.

 

[Taluric]

I guess in the end I don't understand what value there is in a proof that can prove false equations. Unless complex numbers are considered an inherent part of the possible answer set and my instructor simply doesn't want to 'confuse' us with too much information? I dislike being given sets of rules which are only half truths or misrepresentations that we have to correct later.

One way or another I do thank you for the answer given. It'll be interesting to find out what you mean by one sided limits being thrown out the window whenever I get that far.

 

[daon2]

I guess in the end I don't understand what value there is in a proof that can prove false equations. Unless complex numbers are considered an inherent part of the possible answer set and my instructor simply doesn't want to 'confuse' us with too much information? I dislike being given sets of rules which are only half truths or misrepresentations that we have to correct later.

One way or another I do thank you for the answer given. It'll be interesting to find out what you mean by one sided limits being thrown out the window whenever I get that far.

These are not half-truths, though some things are true in greater generality; it is one of the focuses of mathematics to generalize, so that statements may apply to a wider range of structures. The assumption in a typical "Calculus 1" class is that functions have a domain and range, each a subset of the real numbers. Functions can only be continuous where they are defined, and the real-valued square root function is simply not defined at negative real numbers, so "left-hand" limits to 0 make no sense.

One can just as easily talk about continuous functions on the integers, or rational numbers. Pretend for a moment that we agree on some notion of "continuity" of functions from the integers to the integers. Then the function f(n) = (n+1)/2 is not continuous at any even integer, simply because its image is not an integer, and so f is not defined there. Compare this to the square root you have.

Though to understand what continuity really means, you would have to spend some time learning about topological spaces