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lim(x→1)f(x)存在,且f(x)=x^3+sin(x^2-1)/(x-1)+2lim(x→1)f(x),求f(x)=?
题目详情
lim(x→1)f(x)存在,且f(x)=x^3+sin(x^2-1)/(x-1)+2lim(x→1)f(x),求f(x)=?
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答案和解析
lim(x→1)f(x)＝lim(x→1)[x^3+sin(x^2-1)/(x-1)+2lim(x→1)f(x)]
＝1+lim(x→1)[sin(x^2-1)/(x-1)]+2lim(x→1)f(x)
即lim(x→1)f(x)=-1-lim(x→1)[sin(x^2-1)/(x-1)]
而lim(x→1)[sin(x^2-1)/(x-1)]
=lim(x→1){[sin(x^2-1)/(x^2-1)]•(x^2-1)/(x-1)}
=lim(x→1)[(x+1)(x-1)/(x-1)]
=lim(x→1)(x+1)=2
所以 lim(x→1)f(x)=-1-2=-3
f(x)=x^3+sin(x^2-1)/(x-1)-6
＝1+lim(x→1)[sin(x^2-1)/(x-1)]+2lim(x→1)f(x)
即lim(x→1)f(x)=-1-lim(x→1)[sin(x^2-1)/(x-1)]
而lim(x→1)[sin(x^2-1)/(x-1)]
=lim(x→1){[sin(x^2-1)/(x^2-1)]•(x^2-1)/(x-1)}
=lim(x→1)[(x+1)(x-1)/(x-1)]
=lim(x→1)(x+1)=2
所以 lim(x→1)f(x)=-1-2=-3
f(x)=x^3+sin(x^2-1)/(x-1)-6
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