求 lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限.

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/28 19:58:02
求 lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限.

求 lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限.
求 lim(x→∞)[sin(2/x)+cos(1/x)]^x的极限.

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