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


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.

enter image description here

  • $\begingroup$ you are finding the slope of the oblique asymptotes two different ways which one is correct or both correct . oblique asymptote is $y = mx + c$ and how to find the value of c. $\endgroup$ – user120386 Feb 15 '15 at 10:40
  • $\begingroup$ There is one oblique asymptote at $+\infty$ and another at $-\infty$. You find $c$ as $\lim_{t\to\pm\infty}y-m\,x$. $\endgroup$ – Julián Aguirre Feb 15 '15 at 12:07

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 $


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.