Posted by T.Gracken on August 02, 2002 at 17:31:52:
In Reply to: Update on T. Gracken's July 26, 2002 at 12:33:57 post (a limit problem) 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)}.
yes. that was a typo on my part. It is difficult sometimes to check my entered text on the format provided here. sorry
:
: point 2) You are missing a "]" in line (II). Or you
: should not have had any brackets in this line at all?
yes. another typo. although I did not catch it until after posting. (unlike the previous one I just overlooked entirely)
:
: 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)].
no, I used L'Hopital's rule, not the quotient derivative. but I was not about to write every step. the result does lead (provided my earlier typos were corrected
:
: 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.)
thank you. I knew you'd be back.