What is infinite limit of rational trigonometric function?

Question

How can we justify the following limit? $$ \lim _{x \rightarrow -\infty} \dfrac{1+\cos x}{3+2\cos x} $$

We can see the result of limit DNE or $[0, 2]$ on the graph. How can we show?

Answer 1

The given function is bounded. It has as its minimum $0$ and as its maximum $\frac 25$. In fact

$$0=\frac{0}{1}\leq \frac{1+\cos \:x}{3+2\cos \:x}\leq \frac{1+1}{3+2}=\frac{2}{5}=0.4,\quad \forall x \in \Bbb R$$

Hence the limit $$\lim _{x \rightarrow -\infty} \dfrac{1+\cos x}{3+2\cos x}$$ not exist.