ECON 715 Midterm: ECON715 Midterm2Fall2015solution
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Instructions: one double-sided sheet with any content is allowed, calculators are not allowed, show all the calculations, if you need more space, use the back of the page, fully label all graphs. Prove that lim x!0 ln (1 + x) x. This is a limit of the form 0. Using l(cid:146)h(cid:244)pital(cid:146)s rule, ln (1 + x) x. Remark: this is an important result in (cid:133)nance. If interest compounds every instant, the amount can be approximated by er - continuously compounded formula. We can rewrite the required function in a way that allows the use of substitution rule and l(cid:146)h(cid:244)pital(cid:146)s rule: (cid:16)1 + r n(cid:17)n. = exp(cid:16)n ln(cid:16)1 + r n(cid:17)(cid:17) = exp r r=n ! = exp(cid:18)r n(cid:1) ln(cid:0)1 + r ln (1 + x) x (cid:19) where x = r=n. Note that lim n!1 x = 0 lim x!0 ln (1 + x) x. Thus, the required limit can be written as lim n!1(cid:16)1 + r n(cid:17)n.
