Finding the limit of (x + e ^ 2x) ^ - 1 / X when Lim x tends to zero by substitution of equivalent infinitesimal factors

Finding the limit of (x + e ^ 2x) ^ - 1 / X when Lim x tends to zero by substitution of equivalent infinitesimal factors


e^x-1~x,
——》e^x~x+1,
——》e^2x~2x+1,
——》limx→0 (x+e^2x)^(-1/x)
=limx→0 (x+2x+1)^(-1/x)
=limx→0 [(1+3x)^(1/3x)]^-3
=e^-3.



How to calculate LIM (x tends to 0) xcos (1 / x)


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Calculate the limit of (1) Lim x → e (xlnx + 2x) (2) Lim x → π / 2 (SiNx / 2cos2x)


(1)
lim(xlnx+2x)=elne+2e=e+2e=3e
(2)
lim[sinx/(2cos2x)]=sin(π/2)/[2cos(2*π/2)]=1/[2*(-1)]=-1/2