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Let us consider the parametric function x=f(t), y=g(t) for finding the asymptote of parallel to the co-ordinate axes. If t→t_{1}, y=g(t)→∞ such that x→a then x=a is an asymptote. Again If t→t_{1}, x=f(t)→∞ such that y→b then y=b is an asymptote.
As for example, x(t)=t^{2}-1 and y(t)=(t^{2}-1)/t. When t→0 then y(t)→∞ or y(t)→-∞ such that x→-1, hence x=-1 is an asymptote of parametric function. The graph of this parametric function and its asymptote is given belows.

Again consider the parametric function x(t)=2e^{t}+3e^{-t} and y(t)=3e^{t}+2e^{-t} for finding oblique asymptote.If t→∞,then the slope=(dy/dt)/(dx/dt)=(3e^{t}-2e^{-t})/(2e^{t}-3e^{-t})=(3-2e^{-2t})/(2-3e^{-2t})→3/2, hence the asymptote will be y=(3/2)x ⇒2y=3x so the parametric equation of this asymptote is x(t)=2t and y(t)=3t.
If t→-∞,then the slope=(dy/dt)/(dx/dt)=(3e^{t}-2e^{-t})/(2e^{t}-3e^{-t})=(3e^{2t}-2)/(2e^{2t}-3)→2/3, hence the asymptote will be y=(2/3)x ⇒3y=2x so the parametric equation of this asymptote is x(t)=3t and y(t)=2t. The graph of parametric function and its asymptotes are given belows.