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.