I was solving some problems Related to asymptotes, I have a question Can we know number of asymptotes of a 1.rational function 2.trignometric function 3. Implicit function
Without calculating them how can we know how many total number of asymptotes function will have? is we guess this from degree of denominator? but in implicit functions we cannot separate y to put in fraction? plz help me
vertical asymptotes correspond to the roots of the denominator; oblique and horizontal asymptotes appear when the degree of the numerator exceeds that of the denominator by $1$ or less.
the asymptotes of the common trigonometric functions are well-know.
horizontal (resp. vertical) asymptotes appear as the possible finite values of $y$ (resp. $x$) when $x$ (resp. $y$) tends to infinity. For oblique asymptotes, you can try to find horizontal asymptotes after a rotation of the coordinates.
E.g.
$$y^2-x^2=1.$$
If you let one of the coordinates tend to infinity, so does the other. Now if we rotate by $\theta$,
$$(y\sin\theta+x\cos\theta)^2-(y\cos\theta-x\sin\theta)^2=(x^2-y^2)\cos2\theta+xy\sin2\theta=1$$
and with $\theta=\dfrac\pi4$,
$$xy=1$$ has the horizontal asymptote $y=0$.