求极限 lim x趋于无穷x^2(1-xsin1/x)

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求极限 lim x趋于无穷x^2(1-xsin1/x)

求极限 lim x趋于无穷x^2(1-xsin1/x)
求极限 lim x趋于无穷x^2(1-xsin1/x)

求极限 lim x趋于无穷x^2(1-xsin1/x)
求极限 x→∞lim{x²[1-xsin(1/x)]}
令1/x=u,则x=1/u,x→∞时u→0.
故原式=u→0lim{[1-(1/u)sinu]/u²}=u→0lim[(u-sinu)/u³]
=u→0lim[(1-cosu)/(3u²)]=u→0lim[(sinu)/6u]=u→0lim[(cosu)/6]=1/6.