Finding f_L as Source Speed Approaches Speed of Sound

[JJBladester]

Homework Statement

Imagine that the source is to the right of the listener, so that the positive reference direction (from the listener to the source) is in the $+\hat{x}$ direction. If the listener is stationary, what value does $f_L$ approach as the source's speed approaches the speed of sound moving to the right?

Homework Equations

$$f_L=f_s\left ( \frac{v+v_L}{v+v_s} \right )$$

Where $v$ is the speed of the sound in the medium,
$v_L$ is the velocity of the listener, and
$v_s$ is the velocity of the source.

The Attempt at a Solution

I'm assuming that the speed of sound, being so large can be seen as $v_s$ approaching infinity.

I think the way to solve this problem is to take the following limit:

$$\lim_{v_s \to \infty }f_s\left ( \frac{v+v_L}{v+v_s} \right )$$

The answer given is that $f_L$ approaches $\frac{1}{2}f_s$.

I'm not sure how to evaluate the limit to get the answer.

 

[JJBladester]

Got it... $v_s \to v$ so the limit is just a simple substitution and the answer makes sense. I misunderstood what "the speed of sound" meant.