Question

求4道极限的解 要完整的解题过程

计算下列各题极限

Answer 1

1 原式=lim(sinx/-x) =lim cosx/-1 (分子分母分别求导)
=-1
2 原式=lim(1+-2/x)^2x=lim[(1+-2/x)^(-x/2)]^(-4)=e^(-4)
3 原式=lim(1+5/3x)^(-x)=lim[(1+5/3x)^(3x/5)]^(-5/3)=e^(-5/3)
4原式=lim (1+3x)^(1/x)=lim[(1+1/(1/3x))^(1/3x)]^3=e^3

Answer 2

1、
lim(x→0－）sinx/|x|
=lim(x→0－）sinx/(-x)
=-lim(x→0－）sinx/x
=-1

2、
lim(x→∞）(1-2/x)^2x
=lim(x→∞）[(1-2/x)^-x/2)]^(-4)
=e^(-4)
3、
lim(x→∞）[(5+3x)/3x]^(-x)
=lim(x→∞）[(1+5/3x]^(-x)
=lim(x→∞）[(1+5/3x)^(3x/5)]^(-5/3)
=e^(-5/3)

4、lim(x→0）(1+3x)^1/x
=lim(x→0）[(1+3x)^(1/3x)]^3
=e^3

Answer 3

1、lim(x→0¯)sinx/|x|
=lim(x→0¯)sinx/(-x)
=-lim(x→0¯)sinx/x
=-1

2、lim(x→∞)(1-2/x)^(2x)
=lim(x→∞)[(1-2/x)^(-x/2)]^(-4)
=e^(-4)
3、lim(x→∞)[(5+3x)/3x]^(-x)
=lim(x→∞)[(1+5/3x]^(-x)
=lim(x→∞)[(1+5/3x)^(3x/5)]^(-5/3)
=e^(-5/3)

4、lim(x→0)(1+3x)^(1/x)
=lim(x→0)[(1+3x)^(1/3x)]³
=e³