If a left-hand and right-hand limit differs, how to evaluate each mathematically w/o looking at graphs/charts?

Answer 1

You would need the equation, and you can use calculus approaches to find the respective limits.

Answer 2

In practice or theoretically? The principle of left-hand and right-hand limits is elementary: [x→c⁺]lim f(x) = [x→c]lim f|{x: x>c}(x) (where f|{x: x>c} is the restriction of f to the set {x: x>c}). IOW [x→c⁺]lim f(x) = L ⇔ (∀ε>0 ∃δ>0: ∀x∈dom(f)∩{x: x>c} 0<|x-c|<δ ⇒ |f(x) - L| < ε). The left-hand limit is defined similarly.

Now, if you're asking about evaluating right-hand and left-hand limits in practice, very often when you are asked to evaluate a one-sided limit of f at c, f is a function that can be written piecewise, as say f(x) = {g(x) when x>c, h(x) when xc, respectively. So basically all you have to do is look at (or in a few cases, write) the relevant piecewise definition, and then find the limits of the corresponding functions given in that definition.

Answer 3

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