12.20 Form the equation of the tangent line to the graph of the function $f(x)=\frac{x^{2}+4}{x-2}$ at the point of its intersection with the y-axis.
12.20 According to formula (12.6), the equation of the tangent is written in the form $y-y_{0}=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)$, where $\left(x_{0} ; y_{0}\right)$ is the point of tangency. The abscissa $x_{0}$ of the point of intersection of the graph with the $O$ axis is equal to zero, and the ordinate $y_{0}=f(0)=-2$; hence, $(0 ;-2)$ is the point of tangency. Further, using formula (12.4) and the table of derivatives, we get

$$
f^{\prime}(x)=\frac{2 x(x-2)-\left(x^{2}+4\right) \cdot 1}{(x-2)^{2}}=\frac{x^{2}-4 x-4}{(x-2)^{2}}
$$

from which $f^{\prime}(0)=-1$. Thus, the desired equation of the tangent has the form

$y-(-2)=-1 \cdot(x-0)$, or $y=-x-2$.