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Background:

In Team Fortress 2, there are two secondary weapons that the 'Heavy' class can choose between. I am trying to figure out what the best way to compare the two weapons mathematically is.

The first weapon is the default Shotgun. It shoots $1.6$ times per second, and has a clip size of $6$. For simplicity, let's assume that it does $60$ damage every time it hits.

The second weapon is called the Family Business. It shoots $15\%$ faster than the regular shotgun, so it shoots $1.84$ times per second. It has a clip size of $8$, and it does $15\%$ less damage than the regular shotgun, so that means it does $51$ damage every time it hits.

Finally, both weapons must be reloaded after their clips are emptied. It takes $3.5$ seconds for the regular shotgun to reload completely, and it takes $4.5$ seconds for the family business to reload completely.

My Calculations (so far):

The family business shoots $15\%$ faster and does $15\%$ less damage per shot.

$$1.15\cdot0.85 = 0.9775$$

So, ignoring clip sizes, the family business should intuitively do about $2.25\%$ less damage than the regular shotgun while both weapons are continuously firing (ignoring reloading times).

The shotgun takes $6/1.6=3.75$ seconds to empty its clip. The family business takes $8/1.84 \approx 4.35$ seconds to empty its clip. That means that if both weapons are shooting for less than or equal to $3.75$ seconds, then the family business should do $2.25\%$ less damage.

My Question:

Which of these two weapons has a greater average damage per second in an indefinitely prolonged battle? In other words, if $D_s(t)$ and $D_{fb}(t)$ are the damage done after $t$ seconds of continuous shooting by the shotgun and family business, respectively, then what is the value of this limit?

$$\lim_{t\to\infty} \frac{D_{fb}(t)}{D_{s}(t)}$$

The answer should represent ratio of the average damage per second of the family business to that of the regular shotgun, over an indefinitely long shootout.

I know the value for $t=3.75$.

$$\frac{D_{fb}(3.75)}{D_{s}(3.75)}=0.9775$$

Any help is appreciated, thank you!