had a question on limits
what does the negative and positive sign mean next to limits
like limit x approaching 2- then x approaching 2+, how would i solve questions with these limits
x -> 2- means “the limit from the left” - it means to plug in numbers really close to two that are less than two, but not exactly two. 2+ is similar, “the limit from the right”, where you plug in numbers really close to two that are greater than two, but not exactly two.
as an example of where they’re useful: define f(x) to be 1 when x>=2, 0 when x<2 as a piecewise function. now, the limit of f(x) as x-> 2- is 0 right, because numbers less than two that are getting really close to two plugged into f have to evaluate to 0 (remember, you’re not plugging in exactly 2, you’re investigating the behaviour of f as x gets close to two). but the limit of f(x) as x->2+ is 1, because numbers that are bigger than two that are getting really close to two plugged into f have to evaluate to 1. so here, the two limits have different values.
a fact also arises from this: if the 2+ and 2- limits are not equal, then the limit as x goes to 2 doesn’t exist, because the limiting behaviour doesn’t agree on both sides. additionally the function is discontinuous at x=2, because there will be a sharp break in the graph (and of course you can use any value to plug into the limits, im just using 2 here because it’s what you mentioned). so the function i defined above is discontinuous at x=2, and the limit of f as x->2 doesn’t exist
