Take a curve ,some curve which $converges$ to a point as one variable( say $x$)tends to $\infty$
$(ie)$ $\lim_{x\to\infty}$ ${y\to L}$
,the value that it approaches (or) converges to ( $y$ value) ,is the value of the asymptote ,the value that the curve tries to reach($y$ reluctantly tries to get to )but never reaches (or) only reaches the value at $\infty$.
Now the equation of Hyperbola is given by
$\frac{x²}{a²}$ - $\frac{y²}{b²}$=1
To find the asymptotes we substitute $\frac{x²}{a²}$-$\frac{y²}{b²}$=0
Why do we do that ?
What is happening here?
Is there any geometrical reasoning for this?
To find the asymptotes we take the RHS of the equation as 0,why so?
The reason is that the projective equation (in homogeneous coordinates) of your hyperbola is $$ \frac{x^2}{a^2}-\frac{y^2}{b^2}=z^2 $$ and the asymptotes correspond with the points "at infinity", that is, $z=0$.