Prove that `lim_(xto0)(1+3x)^(3/x)=e^(9)`

lim_(xto0)(1+x)^(1/(3x))

Prove that: lim_(xto0)log(1+2x)/(sin3x)=(2)/(3)

Prove that lim_(xto0)((sinx)/(x))=1 (x being in radians ) and hence Show that lim(x to 0) ((tan x)/(x)) = 1 .

Using the epsilon- delta definition prove that lim_(xto0)(2x+3) = 3

Slove lim_(xto0)((1+x)^(1//x)-e)/x

Slove lim_(xto0)((1+x)^(1//x)-e)/x

Slove lim_(xto0)((1+x)^(1//x)-e)/x

Slove lim_(xto0)((1+x)^(1//x)-e)/x

lim_(xto0+)(e^x+x)^((1)/(x))

If f(x)=sgn(x)" and "g(x)=x^(3) ,then prove that lim_(xto0) f(x).g(x) exists though lim_(xto0) f(x) does not exist.