國立政治大學112學年度碩士班暨碩士在職專班
考試科目:微積分
系所別:風險管理與保險系精算科學組
解答:(a)limx→0∫x20t√1+t2dtx4=limx→0(∫x20t√1+t2dt)′(x4)′=limx→0x2√1+x4⋅2x4x3=limx→024√1+x4=12(b)L=(ex−1)1lnx⇒lnL=ln(ex−1)lnx⇒limx→0+lnL=limx→0+(ln(ex−1))′(lnx)′=limx→0+xexex−1=limx→0+(xex)′(ex−1)′=limx→0+xex+exex=limx→0+(x+1)=1⇒limx→0+L=e1=e(c)L=(1+1x)x⇒lnL=xln(1+1x)⇒limx→−∞lnL=limx→−∞ln(1+1x)1x=limx→−∞(ln(1+1x))′(1x)′=limx→−∞−1x(x+1)−1x2=limx→−∞xx+1=1⇒limx→−∞L=e1=e(d)limn→∞(1n+1+1n+2+⋯+1n+2n)=limn→∞n∑k=11n+k=limn→∞n∑k=11/n1+k/n=∫1011+xdx=ln2
解答:(a)By the triangle inequality, we have ||x|−|y||≤|x−y|. Now, limx→af(x)=L. That is, ∀ϵ>0,∃δ>0 such that |x−a|<δ⇒|f(x)−L|<ϵ⇒||f(x)|−|L||≤|f(x)−L|<ϵ⇒limx→a|f(x)|=|L|,Q.E.D.(b)An reverse example: Let f(x)={2x≥a−2x<a, then we have limx→a|f(x)|=|2|, but limx→af(x) does not exist
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解題僅供參考,其他歷年試題及詳解
第三題(a),x^4羅畢達後錯了是4x^3,所以極限不對
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