MATH 1110 Midterm: MATH 1110 Cornell Midterm Solution

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turquoisegnu128

31 Jan 2019

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July 15, 2015: compute the following limits, justifying your steps. Sol: since ex is continuous in r and 1 x + cos(x) is continuous at x = , 1 x +cos(x) = e e lim x ( 1 x +cos(x)) lim x . Sol: let y = x, so x = y, and x is equivalent to y . lim x . Sol: let h = x , then x = + h, and x is equivalent to h 0. lim x . , using the quotient rule. d dx (cid:16)xcos(x) + ln(x)(cid:17) = d dx (cid:16)xcos(x)(cid:17) + To nd d dx (cid:0)xcos(x)(cid:1) we use logarithmic di erentiation. 1 y dy dx cos(x) x sin(x) ln(x) and therefore dy dx. = y(cid:18) cos(x) x sin(x) ln(x)(cid:19) = xcos(x)(cid:18) cos(x) x sin(x) ln(x)(cid:19) Therefore d dx (cid:16)xcos(x) + ln(x)(cid:17) = xcos(x)(cid:18) cos(x) x sin(x) ln(x)(cid:19) +

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