Find the asymptote of the hyperbolic spiral rho=dfrac
Find the asymptote of the hyperbolic spiral $\rho=\dfrac a\varphi$, $\varphi>0$.
Explanation of Method
As the angle shrinks to zero the radius blows up, but the height settles onto a fixed horizontal line.
Step 1: Cartesian coordinates
\[ x=\rho\cos\varphi=\frac{a\cos\varphi}{\varphi}, \qquad y=\rho\sin\varphi=\frac{a\sin\varphi}{\varphi}. \tag{1} \]
Step 2: Behavior as $\varphi\to0^+$
Here $\rho=a/\varphi\to+\infty$, so the spiral recedes to infinity. Examining the coordinates separately: \[\lim_{\varphi\to0^+}x=\lim_{\varphi\to0^+}\frac{a\cos\varphi}{\varphi}=+\infty, \tag{2}\]
\[ \lim_{\varphi\to0^+}y=a\lim_{\varphi\to0^+}\frac{\sin\varphi}{\varphi}=a\cdot1=a. \tag{3} \]
Step 3: Verification
\[a=1,\ \varphi=0.001\text{: }x\approx1000.0,\ y\approx0.99999983\to1\ \checkmark\]
Result
Final answer:
\[\boxed{\,y=a\,}\]
As $\varphi\to0^+$, $x\to+\infty$ while $y\to a$ (finite): the curve approaches the horizontal line $y=a$.