Continuity of $\arctan\big( \frac{ln(2-x)}{(x-2)}e^x \big)$

Question

I have to examen the continuity of this function:

with a an element of [0, +∞[

So far, I've found this:

Using the basic algebraic functions I can rewrite f(x) as:

f(x) = arctg o P o (exp, Q o (ln o T(2) o T, T(-2)))

Now we proof the continuity using the chain rule:

To proof that Q o (ln o T(2) o T, T(-2)) is continuous over ]-∞, 2[ we have to proof that:

1. ln o T(2) o T is continuous over ]-∞, 2[
2. T(-2) is continuous over ]-∞, 2[
3. 0 ∉ T(-2)(]-∞, 2[)

I've been able to proof that 1, 2 and 3 are correct. So for 1 I find that the codomain is R and for 2 that the codomain is ]-∞, 0[.

Now I'm stuck at this point. How can I go on about proving the continuity of this function further on?