Posted by Dr. Andrew Coultos on August 02, 2002 at 13:57:47:
T. Gracken,
let me identify four of your lines in your post by
using I, II, III, and IV. My versions of your lines should be equivalent to yours.
I: lim(x->0){e^ln[e^x + x]^(2/x)}
II: lim(x->0){e^[2ln(e^x + x)/x}
III: lim(x->0){e^[2(e^x + 1)/(e^x + x)]}
IV: e^4
point 1) (I) does not lead to (II). If you move the "2" down as a multiplier on the ln expression, then (I) would lead to this:
lim(x->0){e^[2ln(e^x + x)^(1/x)}.
point 2) You are missing a "]" in line (II). Or you
should not have had any brackets in this line at all?
point 3) (II) does not lead to (III). If your line (II) had been correct, then your line (III) is wrong
because it is not the limit of the derivative seen in
(II). The derivative in line (II) is more complicated
than what you typed for (III), and it is
(e^x + x)[2x(e^x + 1)/(e^x + x) - 2ln(e^x + x)].
point 4) Although (IV), the correct answer, does follow from (III), the work of showing what value the limit is, as a whole, is wrong because (III) as already mentioned has come about by a wrong method.
(If you want, print this off and hold my feet to the fire for any of the particulars of which I discussed.)