Question

How do you evaluate `lim (1+3x)^(4/x)` as x approaches 0?

Answer 1

`lim_(x->0) (1+3x)^(4/x) = e^12`

Explanation:

Note that:

`lim_(t->0) (1+t)^(1/t) = lim_(n->oo) (1+1/n)^n = e`

Using `t = 3x`, we have:

`lim_(x->0) (1+3x)^(4/x) = lim_(t->0) ((1+t)^(1/t))^12`

`color(white)(lim_(x->0) (1+3x)^(4/x)) = (lim_(t->0) (1+t)^(1/t))^12`

`color(white)(lim_(x->0) (1+3x)^(4/x)) = e^12`