この問題を解いてください

質問

次の関数の極限値を求めなさい。
1.lim(x→0)(1-cosx)/x^2
2.lim(x→0)arctanx/x
3.lim(x→π)sinx/(π-x)
4.lim(x→0) (sinx-tanx)/x^3
5.lim(x→0){e^x+e^(-x)-2}/x^2

6.lim(y→0)(1+a/y^2)^y
7.lim(n→0)n{a^(1/n)-1} (a>1) nは整数

よろしくお願いします！

回答No.1

muturajcp

1.
lim_{x→0}(1-cosx)/x^2
=lim_{x→0}(1/2)[{sin(x/2)}/(x/2)]^2
=1/2

2.
lim_{x→0}(arctanx)/x
=lim_{t→0}t/tant
=lim_{t→0}(cost)t/sint
=1

3.
lim_{x→π}(sinx)/(π-x)
=lim_{t→0}(sin(π-t))/t
=lim_{t→0}(sint)/t
=1

4.
lim_{x→0}(sinx-tanx)/x^3
=lim_{x→0}sinx(cosx-1)/(x^3cosx)
=lim_{x→0}sinx(-2{sin(x/2)}^2)/(x^3cosx)
=lim_{x→0}(-1/2){(sinx)/x}[{sin(x/2)}/(x/2)]^2/cosx
=-1/2

5.
lim_{x→0}{e^x+e^{-x}-2}/x^2
=lim_{x→0}[(e^{x/2}-e^{-x/2})/x]^2
=lim_{x→0}[(Σ_{n=0～∞}[(x/2)^{2n}]/(2n+1)!)]^2
=1

6.
lim_{y→0}(1+a/y^2)^y
=lim_{t→∞}e^([a/{(e^t-1)/t^2}]^{1/2})
=1

7.
a>1
lim_{n→∞}n(a^{1/n}-1)
=lim_{t→0}(e^{tloga}-1)/t
=lim_{t→0}Σ_{k=1～∞}t^{k-1}(loga)^k/k!
=loga

lim_{x→0}x(a^{1/x}-1)
=lim_{t→∞}(e^{tloga}-1)/t
=lim_{t→∞}Σ_{k=1～∞}t^{k-1}(loga)^k/k!
=∞