lmits of function

tung

2012-03-09 12:10 am

given that lim x->0(1+x)^1/x=e , find the following limits

1.lim x->0 (1-x)^1/x

2. lim x->0 (1+3x)^1/x

3. lim x->∞ (1+a/x)^x

4. lim x->0 (e^x-1)/x let e^x -1=h

回答 (1)

螞蟻雄兵

2012-03-09 12:51 am

✔ 最佳答案

given that lim(x->0)_(1+x)^(1/x)=e，find the following limits
1.lim(x->0)_(1－x)^(1/x)
Set y=－x
lim(x->0)_(1－x)^(1/x)
=lim(－y->0)_(1+y)^(1/－y)
=1/lim(y->0)_(1+y)^(1/y)
=1/e

2. lim(x->0)_(1+3x)^(1/x)
Set 3x=y
1/(3x)=1/y
1/x=3/y
lim(x->0)_(1+3x)^(1/x)
=lim(y->0)_(1+y)^(3/y)
=[lim(y->0)_(1+y)^(1/y)]^3
=e^3

3. lim(x->∞)_(1+a/x)^x
Set y=a/x
x=a/y
lim(x->∞)_(1+a/x)^x
=lim(y->0)_(1+y)^(a/y)
=[lim(y->0)_(1+y)^(1/x)]^a
=e^a

4. lim(x->0)_(e^x－1)/x
e^x=1+x+x^2/2!+x^3/3!+…
e^x－1=x+x^2/2!+x^3/3!+…
lim(x->0)_(e^x－1)/x
=lim(x->0)_(1+x/2!+x^2/3!+…)
=1

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