first Lim is Pi btw

Every day this site is becoming more cancerous

2)
cos(sqrt(x))^(3/x)
= exp((3/x)*ln(cos(sqrt(x))))
= exp((3/x)*ln(1-x/2 + o(x)))
= exp((3/x)*(-x/2)+o(x))
lim (x ->0) = exp(-3/2)

>>49750

In the third one, replace tan x with its Taylor polynomial approximation: tan x = x + x^3 / 3 + ..., so tan x / x = 1 + x^2 / 3 + ..., and replacing 1/x^2 with n, you get (1 + (1/3) * (1/n))^n, the limit of which is e^(1/3)