Answer
The asymptote of the folium is the line $x+y=-a$ as is shown in Figure 24.
Work Step by Step
From the result in Exercise 81 (a) we obtain the parametrization of the folium of Descartes: $c\left( t \right) = \left( {\frac{{3at}}{{1 + {t^3}}},\frac{{3a{t^2}}}{{1 + {t^3}}}} \right)$.
Note that $t=-1$ is a point of discontinuity of the parametrization.
Figure 24 suggests that the asymptote of the folium is a line of the form $(x+y)$. So, it is natural to find the limit of $(x+y)$ as $t$ approaches -1.
$\mathop {\lim }\limits_{t \to - 1} {\mkern 1mu} (x + y) = \mathop {\lim }\limits_{t \to - 1} {\mkern 1mu} \frac{{3at+3a{t^2}}}{{1+{t^3}}}$
Using the L'Hôpital's Rule twice the limit becomes
$\mathop {\lim }\limits_{t \to - 1} \left( {x + y} \right)$
$= \mathop {\lim }\limits_{t \to - 1} \left( {\frac{{3at + 3a{t^2}}}{{1 + {t^3}}}} \right)$
$= \mathop {\lim }\limits_{t \to - 1} \left( {\frac{{3a + 6at}}{{3{t^2}}}} \right)$
$= \mathop {\lim }\limits_{t \to - 1} \left( {\frac{{6a}}{{6t}}} \right)$
$= \mathop {\lim }\limits_{t \to - 1} \left( {\frac{a}{t}} \right)$
Now, we evaluate this limit as $t$ approaches -1 from the left and from the right:
$\mathop {\lim }\limits_{t \to - 1 - } \left( {x + y} \right) = \mathop {\lim }\limits_{t \to - 1 - } \left( {\frac{a}{t}} \right) = - a$
$\mathop {\lim }\limits_{t \to - 1 + } \left( {x + y} \right) = \mathop {\lim }\limits_{t \to - 1 + } \left( {\frac{a}{t}} \right) = - a$
So, the equation $x+y=-a$ as $t$ approaches $-1$ is a line, where the slope is -1.
Next, we confirm our results by evaluating the slope of the tangent line of $c\left( t \right) = \left( {\frac{{3at}}{{1 + {t^3}}},\frac{{3a{t^2}}}{{1 + {t^3}}}} \right)$ as $t$ approaches $-1$.
Using Eq. (8) the slope of the tangent line is
$\frac{{dy}}{{dx}} = \frac{{y'\left( t \right)}}{{x'\left( t \right)}} = \frac{{6at - 3a{t^4}}}{{3a - 6a{t^3}}}$
So,
$\mathop {\lim }\limits_{t \to - 1 - } \frac{{dy}}{{dx}} = \frac{{ - 9a}}{{9a}} = - 1$
$\mathop {\lim }\limits_{t \to - 1 + } \frac{{dy}}{{dx}} = \frac{{ - 9a}}{{9a}} = - 1$
Hence, we conclude that the asymptote of the folium is the line $x+y=-a$.
