 # (iii) Prove that for each X E (9 b) o lim f'x, = lim f'x) X = f'...

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(iii) Prove that for each X E (9 b) o lim f'x, = lim f'x) X = f'x, (x) + xjx xtx. and lim f(x) = lim fix) = fox + x 1 x. xtx. (iv) Let x be a point in (ab). Prove that (fis differentiable at X ) (fis continuous at x K (fiscontinuous. at x ) (vi) Early in the course last term, we showed that for each real number B > 0 and integer pal, there exists a unique positive real number x such that XP = B. We then used that result to show that for each real number B> there exists a unique monotone increasing function g: R (0,00) such that g(1)=B = and, for all X, YETR glxty) = georgly). Why does it follow at once that g is everywhere differentiable, and that, for all XE TR, (We usually write B* gix, = gengged? for glx, of course. )

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