求极限练习题(二).pdf
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求极限相关练习题(二)
五、n 项和（积）求极限：
1 1 + 2 1 + 2 + 3 + ⋯+ n
1. lim ( 3 + 3 + ⋯+ 3 ) =
n →∞ n n n
1 2 n −1 n
2. lim ( 2 + 2 + ⋯+ 2 + 2) =
n →∞ n n n n
1 + 3 + 5 + ⋯+ (2n −1)
3. lim =
n →∞ 2 + 4 + 6 + ⋯+ 2n
4 8 2n
4. lim ( 2 ∙ 2 ∙ 2 ⋯ 2) =
√ √ √ √
n →∞
1 1 1
5. lim � + 2 + ⋯+ n �=
n →∞ 2 2 2
2 n
x + x + ⋯+ x −n
6. lim =
x →1 x −1
n
7. lim =
n
n →∞ √n!
1 1 1
8. lim � + + ⋯+ �=
n →∞ n + 1 n + 2 n + n
1 1 1
9. lim (1 + 2) × �1 + 4 �× ⋯× �1 + 2n �=
n →∞ 2 2 2
1 22 n2
10. lim ( + + ⋯+ ) =
n →∞ 6 6 6 2
√n + n √n + 2n √n + n
六、利用洛比达法则求极限：
ex x μ
1. lim = 2. lim = (其中μ 0)
x →∞x + 1 x →+∞lnx
x −arctanx
sinx −sina 4. lim =
3. lim = x →0 tanx −x
x →a x −a
lnsinx
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