The limit can be found if I use a calculator but how do I find it without using one? I tried to use the Taylor' series and this is what I have so far but it doesn't look right:
$${(x-{x^3 \over 3} + o(x^5)) (1 - {x^2\over2} o(x^4))(1 + cos(x) + o(cos(x))^2)\over ln(x)}$$
How do I proceed further?
The first step into finding a limit is plugging in where your variable tends to in the limit (here your variable is $x$ and it is tending to the number $2$) sometimes, it happens that you are lucky and it yields a number instead of getting an indeterminate form.
$\lim \limits_{x \to 2} {sin(x)cos(x)e^{cos(x)}\over ln(x)}$ = ${sin(2)cos(2)e^{cos(2)}\over ln(2)}$
Putting this into a calculator would yield $-0.360078...$, you will never be asked to find the result of such a fraction without a calculator.