Question

这道高等数学的题怎么解？求解题过程，谢谢

Answer 1

解：分享一种解法。x→0+，用无穷小量替换时，将√(x^2+1)用广义二项式一并替换。∵ln[x+√(x^2+1)]=ln[1+x+√(x^2+1)-1]，而x+√(x^2+1)-1=x-1+[1+(1/2)x^2+o(x^2)]=x+(1/2)x^2，∴ln[x+√(x^2+1)]=[x+(1/2)x^2]-(1/2)[x+(1/2)x^2]^2+o(x^2)=x+o(x^2)。故，原式=lim(x→0+)[1/x-1/(x-x^2/2)]=lim(x→0+)[(-1/2)x^2]/[x^2(1-x/2)]=-1/2。供参考。