Question

当x→无穷时,3x-1/3x+1的2x次方的极限

Answer 1

问：当x→无穷时,3x-1/3x+1的2x次方的极限

答：稍等哦

答：lim(x->∞)[(3x-1)/(3x+1)]^(2x+1)=lim(x->∞){[(3x+1)-2]/(3x+1)}^(2x+1)=lim(x->∞)[1-2/(3x+1)]^(2x+1)=lim(x->∞)[1-2/(3x+1)]^[-(3x+1)/2]*[-2(2x+1)/(3x+1)]=lim(x->∞)e^[-2(2x+1)/(3x+1)]=e^lim(x->∞)[-2(2x+1)/(3x+1)]=e^lim(x->∞)[(-4x-2)/(3x+1)]=e^[(-4-0)/(3+0)]=e^(-4/3)

答：lim(x/(x+2))的3x次方，x趋向于无穷设t=(x/(x+2))^3x，则lnt=3xln(x/(x+2))1/3lim{lnt}=lim{xln[x/(x+2)]}=lim{ln[1-2/(x+2)]/[1/x]} (化为0/0型未定式，用洛必达法则)=lim{[(x+2)/x*2/(x+2)^2]/[-1/x^2]}=lim{[-2/(x+2)]}=0即lim{lnt}=0∴limt=lim(x/(x+2))=1lim趋向于无穷大时（e的x/2次方）-1的极限=lim x分之2=0当x趋向于无穷大时，（2/3）的x次方，求极限？lim(x-->∞）（2/3）^x=lim(x-->∞）1/（3/2）^x=0求极限 lim(xsin2/x+sin3x/x) x趋向于无穷lim(xsin2/x+sin3x/x) = lim(xsin2/x)+limsin3x/xx趋向无穷大2/x趋向于0，使用等价无穷小替换lim(xsin2/x)=2，又因为sin3x有界而x为无穷大，有界/无穷大=0，所以lim(xsin2/x+sin3x/x) =2lim（x趋向于无穷大）（5x^3-3x+1） 求极限因为lim（x趋向于无穷大）1/（5x^3-3x+1）=lim（x趋向于无穷大）(1/x^3)/（5-3/x^2+1/x^3）=0所以原式=∞。arctan(x^2)/x的极限当x趋向于无穷当x→∞时，由影象可以看出arctan(x^2)→π/2 (即tant的反函式影象，其中该处的t为x^2)又∵当x→∞时，1/x→0故此题为无穷小×有界函式的问题∴lim(x→∞)arctan(x^2)/x=0