Real analysis: is there a simple way to get an upper and lower limit for the function \$ f (x) = frac { sin ( sin {(x)} + x)} {2+ cos {( lvert x rvert} + cos {(x)})} \$

Leave $$f: mathbb {R} rightarrow mathbb {R}$$

$$f (x) = frac { sin ( sin {(x)} + x)} {2+ cos {( lvert x rvert} + cos {(x)})}$$

I want to check the lower and upper limits. Here is what I was thinking:

The image / range of $$without {(x)}$$ Y $$cos {(x)}$$ is $$[-1,1]$$. Therefore, assuming $$without {x} = 1$$ Y $$cos {x} = – 1$$ I get: $$f_ {max} = frac {1} {2-1} = 1$$

The other possibility is $$without {x} = – 1$$ Y $$cos {x} = – 1$$:

$$f_ {min} = frac {-1} {2-1} = – 1$$

So it seems to me that $$f (x)$$ I would never have values ​​outside $$[-1,1]$$. However, while that interval could be a lower and upper limit, I would like to be a little more specific and determine the maximum / minimum or the minimum / maximum of $$f (x)$$. Any idea how I would do that?