Verify if the limit of the function arctan3xarctan5x exists

Verify if the limit of the function arctan(3x)/arctan(5x) exists if x goes to 0?

Solution

We\'ll recall the remarcable limit formula:

lim arctan(u(x))/u(x) = 1, if x approaches to 0.

We\'ll create the remarcable limits in the given function:

lim [3x*(arctan 3x)/3x]*[(5x)/5x*(arctan 5x)] = lim 3x*lim [arctan (3x)/3x]*lim[(5x)/arctan (5x)]*lim (1/5x)

By definition, lim [arctan (3x)/3x] = 1 and lim[(5x)/arctan (5x)] = 1

lim [3x*(arctan 3x)/3x]*[(5x)/5x*(arctan 5x)] = lim 3x/*lim (1/5x)

lim [3x*(arctan 3x)/3x]*[(5x)/5x*(arctan 5x)] = (3/5)*lim(x/x)

The limit of the given function, when x approaches to 0, is : lim arctan(3x)/arctan(5x) =3/5.