求极限 lim(x趋向正无穷)(e^x+x)^(1/x)

问题描述:

求极限 lim(x趋向正无穷)(e^x+x)^(1/x)

用洛必达法则:原式=lim(x→+∞)e^(ln(e^x+x)/x)=lim(x→+∞)e^((e^x+1)/(e^x+x)/1)=lim(x→+∞)e^((e^x+1)/(e^x+x))=lim(x→+∞)e^(e^x/(e^x+1))=lim(x→+∞)e^(1/(1+1/e^x))=e^1=e