로피탈 정리

First, we note that since both the numerator and denominator of $R$ are quadratic, the limit as $x \to \infty$ can be found by taking the quotient of the leading coefficients. In particular,

$$\lim_{x \to \infty} R(x) = \frac{3}{5}$$

and thus the horizontal asymptote of $R$ is $y = \frac{3}{5}$.

Next, since $b$ and $c$ are distinct, we see that the numerator of $R$ is zero (and the denominator is not) when $x = b$, and thus $R(b) = 0$. In addition, the denominator is zero (and the numerator is not) when $x = c$, and this results in there being a vertical asymptote at $x = c$.

Finally, we observe that $x = a$ makes both the numerator and denominator zero. Since $R(a)$ is thus not defined, the graph of $R$ has either a hole or a vertical asymptote at $x = a$. To determine which, we evaluate the limit and see that

$$\lim_{x \to a} \frac{3(x-a)(x-b)}{5(x-a)(x-c)} = \lim_{x \to a} \frac{3(x-b)}{5(x-c)} = \frac{3(a-b)}{5(a-c)}\,.$$ This tells us that $R$ has a hole at the point $(a, \frac{3(a-b)}{5(a-c)})$. Moreover, $R$ is not continuous at $x = a$ and $x = c$.

A possible graph of $R$ is shown in the following figure.