MATH 171 Final: Math 171 TAMU 171Fall 15 FinalExamolutions
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However, this leads to trying to nd the limit as x goes to in nity of. Note that this ratio has the form / , so we should be able to use l"hopital"s rule. 12 cos x sin x 30. This limit does not exist, which tempts one to say the original limit does not exist. This is not a valid conclusion to make, as this problem shows. L"hopital"s rule is not an if and only if result. (b) lim x (cid:18)1 + 2 x(cid:19)x lim x (x ln(1 + 2/x)) = lim x (cid:18) ln(1 + 2/x) 2 x(cid:19)x lim x (cid:18)1 + x 0(cid:18)sin 1(x) (cid:19), x (c) lim. = lim x (cid:0)ex ln(1+2/x)(cid:1) = e2 x 0(cid:18) sin 1(x) lim x (cid:19) = lim x 0(cid:18)(1 x2) 1/2. 1 (cid:19) = 1 lim x 2(cid:0)eln(x+1)(cid:1) = lim x 2 (x + 1) = 3 (d) lim x 2(cid:0)eln(x+1)(cid:1) (e) lim x 0(cid:18)23x 5x.
