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求极限lim(x→∞)（(x+2)/(x+1))^x
题目详情
求极限lim(x→∞) （(x+2)/(x+1))^x
▼优质解答
答案和解析
lim(x->∞) [(x+2)/(x+1)]^x
= lim(x->∞) [((x+1)+1)/(x+1)]^x
= lim(x->∞) [1 + 1/(x+1)]^x
= lim(x->∞) [1 + 1/(x+1)]^[(x+1) * x/(x+1)]
= e^lim(x->∞) x/(x+1)
= e^lim(x->∞) 1/(1+1/x)
= e^[1/(1+0)]
= e
= lim(x->∞) [((x+1)+1)/(x+1)]^x
= lim(x->∞) [1 + 1/(x+1)]^x
= lim(x->∞) [1 + 1/(x+1)]^[(x+1) * x/(x+1)]
= e^lim(x->∞) x/(x+1)
= e^lim(x->∞) 1/(1+1/x)
= e^[1/(1+0)]
= e
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