作业帮lim(X-∞)X[√(x2+1）-x]请问这道极限题怎么算啊
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/26 14:44:21
![lim(X-∞)X[√(x2+1）-x]请问这道极限题怎么算啊](/uploads/image/z/15222564-36-4.jpg?t=lim%28X-%E2%88%9E%29X%5B%E2%88%9A%28x2%2B1%EF%BC%89-x%5D%E8%AF%B7%E9%97%AE%E8%BF%99%E9%81%93%E6%9E%81%E9%99%90%E9%A2%98%E6%80%8E%E4%B9%88%E7%AE%97%E5%95%8A)
lim(X-∞)X[√(x2+1）-x]请问这道极限题怎么算啊
lim(X-∞)X[√(x2+1）-x]
请问这道极限题怎么算啊
lim(X-∞)X[√(x2+1）-x]请问这道极限题怎么算啊
答：
因为：[√(x^2+1)-x ]*[√(x^2+1)+x]=x^2+1-x^2=1
所以：
lim(x→∞) x[√(x^2+1)-x]
=lim(x→∞) x/[√(x^2+1)+x]
=lim(x→∞) x/(x+x)
=1/2
中括号内利用平方差公式写成（X²+1-X)/(根号（X²+1）+X)，再上下约分X，最后用无穷大代入X
分子分母同时乘以 [√(x2+1）+x]，则原式化为x/[√(x2+1）+x],当x->∞时，极限为1/2
lim(x->∞)x[√(x^2+1）-x]
=lim(x->∞)x [√(x^2+1）-x][√(x^2+1）+x]/ [√(x^2+1）+x]
=lim(x->∞) x / [√(x^2+1）+x] (∞/∞)
=lim(x->∞) 1 / (x/√(x^2+1）+1 )
=...
全部展开
lim(x->∞)x[√(x^2+1）-x]
=lim(x->∞)x [√(x^2+1）-x][√(x^2+1）+x]/ [√(x^2+1）+x]
=lim(x->∞) x / [√(x^2+1）+x] (∞/∞)
=lim(x->∞) 1 / (x/√(x^2+1）+1 )
=lim(x->∞) √(x^2+1) / (x+√(x^2+1) )
=lim(x->∞) √(1+1/x^2) / (1+√(1+1/x^2) )
=1/2
收起