課程名稱︰微積分 1 課程性質︰生物環境系統工程學系/工程科學及海洋工程學系/地質科學系 課程教師︰呂治鴻 開課學院：理學院 開課系所︰數學系 考試日期︰2025/09/15(一) 考試時限：50 分鐘 試題 : National Taiwan University - Calculus 1 (Class 09) - Quiz 1 2025/09/15 (Monday) - 50 minutes Name:　　　　　　　　　　　　Student ID Number:　　　　　　 Therefore are FOUR questions in this quiz. Your work is graded on the quality of your writing as well as the validity of the mathematics. Caution: you cannot apply L'Hopital's Rule. 1. (10 points) Evaluate each of the following limits or show that it does not exist. 3 πx (a) (5 points) lim (x-1)sin(x tan -----). x->1 2 [exp(-x)] (b) (5 points) lim (1 + e ), where [s] is defined as the largest x->∞ integer that is less than or equal to s. 2. (20 points) √(ax+b)-3 (a) (8 points) Find real numbers a and b such that lim ------------- = 1 x->0 x (b) (8 points) Let f: |R -> |R be a continuous function defined by ｛ (sin(6x)+a-2b)/(3x) if x ≠ 0; f(x) = ｛ ｛ 2a+b if x = 0. Find the pair (a,b). (c) (4 points) Find the horizontal asymptotes of the function g(x)=√(x^6+3x^3)+x^3. 3. (15 points) sin(x) (a) (3 points) Please use lim -------- = 1 to evaluate x->0 x 1-cos(x) the limit lim ------------. x->0 x^2 x-sin(x) (b) (2 points) Assume that L = lim ----------. x->0 x^3 (x/2) - sin(x/2) Find lim ------------------ in terms of L. x->0 x^3 (c) (10 points) Use (a)-(b) and the identity x-sin(x)=2[(x/2)-sin(x/2)+sin(x/2) (1-cos(x/2))] to evaluate x-sin(x) the limit lim ----------. (You do not need to prove the limit exists.) x->0 x^3 4. (5 points) Assume that f : |R -> |R is differentiable at a∈|R. Recall that f(x)-f(a) f'(a) = lim ----------- . x->a x - a (a) (3 points) Write down the precise definition (ε-δ definition) of f(x)-f(a) lim ----------- = f'(a). x->a x - a (b) (2 points) Note that the identity f(x) - f(a) f(x) = f(a) + ------------- ．(x-a) for x ≠ a. x - a Show that f is continuous at a. --

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