Finding asymptote of specific function with parameter

Question

How can I find the asymptotes and points of rising and falling in the function:

$$\dfrac{\sqrt{a^2-x^2}}{x}$$

Answer 1

Assume that $a>0$. This function is defined on $[-a,0)\cup(0,a]$. Since the domain is bounded, it can only have vertical asymptotes. The vertical asymptote is $x=0$. For $x>0$, as $x$ increase, the denominator increases, while the numerator decreases, so the function is decreasing. Since this is an odd function, it is increasing for $x<0$.