当x趋近于正无穷时,求lim[x+根号（1+x^2)]^1/x的极限

Question

当x趋近于正无穷时,求lim[x+根号（1+x^2)]^1/x的极限

Answer 1

∵lim(x->+∞)[ln(x+√(1+x^2))/x]
=lim(x->+∞)[1/√(1+x^2)] (∞/∞型极限,应用罗比达法则)
=0
∴lim(x->+∞)[(x+√(1+x^2))^(1/x)]
=lim(x->+∞){e^[ln(x+√(1+x^2))/x]}
=e^{lim(x->+∞)[ln(x+√(1+x^2))/x]}
=e^0
=1.