We are assuming that exists. That is, given any , there exists ...

Solution Summary

A differentiable function is continuous at any point for which the limit of the derivative exists. The solution is a step by step proof of that fact comprising 3/4 of a page in Word with equations written in Mathtype. (Although the question is not worded in that way, that is in fact what is being proved) The proof uses the mean value theorem, which is frequently useful in such proofs and so serves as a useful illustration. Also given is an example of a function with a discontinuous derivative.

... 1/x^2))= lim((2/x^3)/exp(1/x^2))= lim(2u^3 ... and observe that the limit is zero and you can assume u=1 ...0 x=0 Show that the n th derivative of f (x) exists for all ...

... b) Set f(x) = (x^2)sin(1/x) for x neq 0 and f...f (an ) f (bn ) = f ′(0) and lim = f ′(0) . lim an bn n ... So for any ε > 0 , we can find some N > 0 , such that ...

... δ > 0 such that |x a| < ε ⇒ |f(x) f(a)| < δ. ... is not continuous at = 0 since lim → satisfies the ... at = 0 since for every > 0, we can choose = 1 ...

... Show that if L=0 then lim as x goes to c of f(x... x_(2n+1)) = -1 - 1/(2n+1) goes to 0, then we can find two ... I think L should be c. The limit of f(x) as x goes to ...

... When x → 1 , both f(x) and g(x) are → 0 , so we can apply L ... 1 ( ln x ) ' = lim x ln x 1 1 lim = lim = lim ( ) x →1 π cos(π x) π x →1 x cos(π x) x...

...lim ym =x m ...lim f (ym ) = 0.i1 i2 i3 · · · ij −1 (ij − 1)999 · · · m ... Next, let zm be the element of (0, 1) that can be expressed as the terminating ...

... Therefore you can choose the format that is most suitable ... hh lim (1.7) lim hh h 0 h 0. ... if a function f(x) is continuous on the closed interval ...

... enclose a rectangular pen adjacent to a long existing wall ... zero, any of the terms in the parenthesis can be zero ... 4 lim f x lim 0 x 6 x x...