Find the asymptotes of the Parametric equation?

Question

Consider $$ x(t) = 2 e^{-t} + 3e^{2t}$$

$$y(t) = 5 e^{-t} + 2 e^{2t}$$

which represents a non rectilinear paths

Horizontal and Verical Asymptotes :

If $t \rightarrow +\infty \ \ or \ \ -\infty$, then $x(t) \ \ and \ \ y(t) \ \ \rightarrow \infty$, So there are no asymptotes parallel to coordinate axis

oblique Asymptotes:

Please tell me how to find the Oblique asymptotes

Answer 1

There are no horizontal asymptotes: this would mean $x\to\infty$ and $y\to$ some finite value. For obligue asymptotes look at the limit when $t\to\pm\infty$ of $y/x$. This is a plot of the curve.

Answer 2

There are two asymptotes by inspection which are at an angle to x-axis.

We need to find out not $ \frac{y(t)}{x(t)} $ tendency but tendency of limits of oblique asymptotes.

$ \dfrac{dy/dt}{ dx/dt}$ when $t\to +\infty $

and also

$ \dfrac{dy/dt}{ dx/dt}$ when $t\to -\infty $

separately.

These limits evaluate to $5/2$ and $2/3$ for each asymptote as coefficients for positive and negative exponents.

Better to make parametric plot of the curve and an ordinary plot of the slope.