MATH 41 Midterm: MATH 41 Stanford Exam1Solutions
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Solutions to math 41 first exam october 14th, 2014: (25 points) compute the following limits. If the limit does not exist, explain why. Since sin2(x) 0, we get the inequalities. 2 sin2(x) sin2(x) cos(1/x) 2 sin2(x), x 6= 0 since limx 0. 2 sin2(x) = 0, by the squeeze theorem we nd (sin2 x)2cos(1/x) = 0. lim x 0 (b) lim t 0(cid:18) We do the following algebraic manipulation: t 0(cid:18) lim. 1 1 + t t 1 + t. 1 t (1 1 + t)(1 + 1 + t) (t 1 + t)(1 + 1 + t) Expanding the numerator we nd the limit equals. 1 (1 + t) t 1 + t(1 + 1 + t) T t 1 + t(1 + 1 + t) The nal expression is continuous at t = 0, so we may evaluate, nding t 0(cid:18) lim.