「Topic:極限」の版間の差分

As x gets closer to 0 the function keeps oscillating between -1 and 1. It is also true that sin(1/x) oscillates an infinite number of times on the interval between 0 and any positive value of ''x''. The sine function, sin(x), is equal to zero whenever x=kπkπ, where k is a positive integer. Between each value of k, sin(x) oscillates between 0 and -1 or 0 and 1. Hence, sin(1/x)=0 for every x=1/(kπkπ). In between consecutive pairs of these values, 1/(kπkπ} and 1/[(k+1)ππ], sin(1/x) oscillates from 0, to -1, to 1 and back to 0 and so on. We may also observe that there are an infinite number of such pairs, and they are all between 0 and 1/ππ. There are a finite number of such pairs between any positive value of ''x'' and 1/ππ which implies that there must be infinitely many between ''x'' and 0. From our reasoning we may conclude that as ''x'' approaches 0 from the right, the function sin(1/x) does not approach any value. We say that the limit as ''x'' approaches 0 from the right does not exist. (In contrast, in the case of a "jump", the limits from each side did exist. Since they weren't equal, though, the regular limit didn't exist.)