Question

请问这几个极限的题目怎么写啊？

Answer 1

(1)
lim[x→0]sin(2x)/x
=2lim[x→0]sin(2x)/(2x)
=2lim[x→0]sin(x)/x
=2

(2)
lim[x→∞](1+2/x)^x
=lim[x→∞]((1+2/x)^(x/2))^2
=lim[x→∞]((1+1/x)^x))^2
=exp(2)

(3)
lim[x→0](1-3x)^(2/x)
=lim[x→0]((1-3x)^(-1/(3x)))(-2/3)
=lim[x→0]((1+x)^x)(-2/3)
=exp(-2/3)

(4)
lim[n→∞]2^nsin(x/2^n) (x≠0)
=lim[n→∞](sin(x/2^n)/(x/2^n))/x
=1/x

(5)
lim[x→0](xsin(1/x)+1/xsinx)
=lim[x→0]xsin(1/x)+lim[x→0]1/xsinx
=0+1
=1

(6)
用洛必达法则
lim[x→0](tanx-sinx)/x^3)
=lim[x→0]((1-cosx)/x^2)(1/cosx)(sinx/x))
=1/2