Epsilon-Delta definition of a Limit: Question

[GeoMike]

This may be a dumb question, but:

Using the epsilon-delta definition of a limit, when x is approaching the maximum of a parabola, like with: $$\lim_{x\rightarrow0}-x^2$$, obviously any value of epsilon puts $$L+\epsilon$$ outside the range of f(x). So, I take it in this case only $$L-\epsilon$$ is considered (which made sense to me graphically since you now have two points where the function intersects this line)?

I just need verification (or correction) on this because every example and problem in the two textbooks I have has given/displayed an L and an epsilon such that $$L-\epsilon$$ and $$L+\epsilon$$ both lie within the range of f(x).

-GeoMike-

 

[fourier jr]

This may be a dumb question, but:

Using the epsilon-delta definition of a limit, when x is approaching the maximum of a parabola, like with: $$\lim_{x\rightarrow0}-x^2$$, obviously any value of epsilon puts $$L+\epsilon$$ outside the range of f(x). So, I take it in this case only $$L-\epsilon$$ is considered?

no. you find the limit at a max/min just like you would at any other point. i think the only instance where the direction you approach a certain valuie matters is when you approcah from the right or left, not from above/below.

 

[0rthodontist]

x always does take values between L + e and L - e for L = 0 and e an arbitrary positive number. The values all happen to be between L and L - e, with none greater than L, but this is extra information that you don't need to evaluate the limit.