高数求极限。第四题
2个回答
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4.解:lim(x->0)[sin(3x)/sin(5x)]
=lim(x->0)[(3/5)*(sin(3x)/(3x))*(sin(5x)/(5x))]
=(3/5)*lim(x->0)[sin(3x)/(3x)]*lim(x->0)[sin(5x)/(5x)]
=(3/5)*1*1 (应用重要极限lim(z->0)(sinz/z)=1)
=3/5。
lim(n->∞)[2^n*sin(x/2^n)]
=lim(n->∞)[x*sin(x/2^n)/(x/2^n)]
=x*lim(n->∞)[sin(x/2^n)/(x/2^n)]
=x*1 (应用重要极限lim(z->0)(sinz/z)=1)
=x。
=lim(x->0)[(3/5)*(sin(3x)/(3x))*(sin(5x)/(5x))]
=(3/5)*lim(x->0)[sin(3x)/(3x)]*lim(x->0)[sin(5x)/(5x)]
=(3/5)*1*1 (应用重要极限lim(z->0)(sinz/z)=1)
=3/5。
lim(n->∞)[2^n*sin(x/2^n)]
=lim(n->∞)[x*sin(x/2^n)/(x/2^n)]
=x*lim(n->∞)[sin(x/2^n)/(x/2^n)]
=x*1 (应用重要极限lim(z->0)(sinz/z)=1)
=x。
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