求 lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限。需要详细步骤。谢谢！

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lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限。需要详细步骤。谢谢！
解答：
lim(x→∞)[sin(2/x)+cos(1/皮郑x)]^x
=lim(x→∞)[1+（sin(2/x)+cos(1/x)-1）]^x
=lim(x→∞)[1+（sin(2/x)+cos(1/x)-1）]^[1/（（sin(2/x)+cos(1/x)-1））]*[x*（燃晌颂（sin(2/x)+cos(1/谨悉x)-1））]
=e^lim[x*（（sin(2/x)+cos(1/x)-1））]
=e^lim(t→0)[（（sin(2t)+cos(t)-1））/t]
=e^lim(t→0)[2（sin(2t)/2t+(cos(t)-1）/t)]
=e^(2-0)=e^2

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