deltas for limits
For a given epsilon e > 0, I need to find a delta d such that for all x, abs(x) < d.
I have these three:
abs(sqrt(x+4) - 2) < e
For this, I rationalized abs(sqrt(x+4) - 2) and got abs(x/(sqrt(x+4)+2), which I said is <= abs(x/2). Then I chose delta to be e/2. Is this ok?
The second one is abs(((1-x)^4) - 1) < e and the third one is abs([(x^2 - x + 1)/(x+1)] - 1) < e. I'm not really sure how to do these. Could someone show me how to find the appropriate d(e)? I'm somewhat lost at the moment and I have a whole bunch to solve.
Can you quickly show me how to do the other two? I have to do like ten and some of them look similar to these, so if I could see how they are done, I should be ok. Also, I don't really see where this is coming from exactly. We're learning about the delta-epsilon notation of limits of functions, however there's no accumulation point c here for 0 < abs(x-c) < d(e). So how exactly are you picking the deltas?
If then, (given that ).
Thanks for the help man.
For the last one, I get the absolute value of:
x^2 - 2x
--------, which is less than or equal to the absolute value of
x + 1
x^2 - 2x
--------, which equals the absolute value
x-2, which is less than or equal to
abs(x) + 2.
How do I then find the delta for this?