The discussion centers around the limit expression lim_{Δx → 0} f(x + Δx) · Δx and whether it equals zero. Participants explore conditions under which this limit may hold true, including considerations of continuity and boundedness of the function f.
f is continuous at x or if lim_{Δx → 0} f(x + Δx) exists.f is bounded near x, the limit can be shown to equal zero using the Squeeze Theorem.f is unbounded near x, leading to the possibility that the limit may not exist or may not equal zero.lim_{Δx → 0} f(x_0 + Δx) · Δx = k for some constant k, seeking further conditions on f.f(x) = 1/x, to illustrate cases where the limit does not equal zero.Participants generally agree that the limit does not hold true in all cases, with multiple competing views on the conditions under which it may or may not equal zero. The discussion remains unresolved regarding specific functions that may contradict the general statement.
Limitations include the dependence on the continuity and boundedness of the function f, as well as the specific behavior of f near the point of interest. The discussion does not resolve the implications of these conditions.
gabel said:Is the following true, if no is there som theory i can studdy?
##\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0##
Who told you that? gopher_p, in the only other response here, said "this is not always true".gabel said:I was told in general, so there must be some functions that does the oppeist?