Example 2. If $x$ is a real number, find the maximum and minimum values of $\frac{3-\sin x}{4-2 \cos x}$.
The objective function can be seen as the slope k of the line connecting any point $(2 \cos x, \sin x)$ on the ellipse $-\frac{x^{2}}{4}+y^{2}=1$ and the fixed point $(4,3)$.
From the graph, it is easy to see: when the line
$y=k(x-4)$
+3 is tangent to the ellipse,
k reaches its extreme values.
Using formula 2.2
we get: $4 k^{2}+1$
$=(3-4 k)^{2}$, which simplifies to $3 k^{2}-6 k+2=0$, solving this gives
$\mathrm{k}=1 \pm \frac{\sqrt{3}}{3} \therefore \frac{3-\sin \mathrm{x}}{4-2 \cos \pi}$ has the maximum value of
$1+\frac{\overline{3}}{3}$, and the minimum value is $1-\frac{\sqrt{3}}{3}$.