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lim[(x-1)/(x+1)]^(x/2+4)求极限x→∞
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lim[(x-1)/(x+1)]^(x/2+4)求极限x→∞
▼优质解答
答案和解析
lim [(x-1) / (x+1)]^(x/2 + 4)
= lim [(x+1-2) / (x+1)]^(x/2 + 4)
= lim [1 - 2/(x+1)]^(x/2 + 4)
= lim {1 + 1 / [-(x+1)/2]} ^ {-(x+1)/2 * -2/(x+1) * (x/2 + 4)]
= e^(-2)lim (x/2 + 4) / (x+1),公式lim(x->∞) (1+1/x)^x = e,这里x = -(x+1)/2
= e^(-2)lim [(1/2 + 4/x) / (1+1/x)]
= e^(-2) * [(1/2+0)/(1+0)]
= e^(-1)
= 1 / e
= lim [(x+1-2) / (x+1)]^(x/2 + 4)
= lim [1 - 2/(x+1)]^(x/2 + 4)
= lim {1 + 1 / [-(x+1)/2]} ^ {-(x+1)/2 * -2/(x+1) * (x/2 + 4)]
= e^(-2)lim (x/2 + 4) / (x+1),公式lim(x->∞) (1+1/x)^x = e,这里x = -(x+1)/2
= e^(-2)lim [(1/2 + 4/x) / (1+1/x)]
= e^(-2) * [(1/2+0)/(1+0)]
= e^(-1)
= 1 / e
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