MATH 1A Final: MATH 1A Harvard 1a Final 2004 Spring Solutions
Document Summary
Full credit may not be given for an answer alone. You may use the backs of the pages or the extra pages for scratch work. Students who, for whatever reason, submit work not their own will ordinarily be required to withdraw from the college. 1: (15 points) find the following limits. (i) lim x . Since this limit is of the form . H (ii) lim x 2 x2 4 x2 + 4. This limit is not of an indeterminate form, since the numerator goes to 0 whilst the denominator goes to 8. Hence by the ordinary limit laws, the limit is 0. Using l"h opital"s rule gives an answer of 1, which is incorrect. (iii) lim x 0+ (cos x)1/x. This limit is of the form 1 , which is indeterminate. Hence we (cos x)1/x = l, we have can use l"h opital"s rule on its logarithm. = lim x 0+ ln(cos x)1/x ln(cos x) x.
