Question

How do you find the limit of `(cos (2x))^(3/(x^2))` as x approaches 0?

Answer 1

The limit should be `1/e^6`.
`lim_(x->0)cos^(3/x^2)(2x)=`

But:
`cos^(3/x^2)(2x)=e^[3/x^2ln[cos(2x)]` (have a look at the properties of logarithms)

and:
`lim_(x->0)e^[3/x^2ln[cos(2x)])=e^-6`

The exponent `3/x^2ln[cos(2x)]` tends to `-6`:

hope it is clear