Suppose you travel a distance d (in miles) at an average speed of 20 mph and travel back at x mph. Explain why your average speed is 2d/...

1)

The distance travelled 
d+d = 2d

The time taken for forward jpurney =
d/20

The time taken for the return journey =
d/x.

So the total time taken for up and down =
d/20+d/x.

Therefore the average speed = total distance/
total time taken = 2d/{d/20+d/x}
mph..

 2)

2d/{d/20+d/x) mph =
2d / d{(x+20)/20x } mph.

2d/{d/20+d/x)mph = 2*20x/(x+20)
mph.

2d/{d/20+d/x) mph= 40x/(20+x)
mph.

So 40x/(20+x) is the saverage in simplified rational
form in the lowest
terms.

3)

If f(x) = 40x/(20+x)
, then to find the limit as x-->infinity.

Since both
numerator and denomonator become infinite,  f(x) becomes indeterminate as x
goes infinite. So we divide both numerator and denominator by x and then take the
limits.

f(x) = (40x/x){20/x+x/x} =
40/{20/x+1}

Therefore, Lt x--> infinity  f(x) =
40/(0+1} = 40 mph.

Practical interpretation: The time taken
to forward journey = d/20. The time taken for return journey is d/x is zero as the speed
x is  very very high. So the up and down journey distance 2d  took only  d/20 hours of
time. So the average speed  is 2d/ (d/20) = 40d/d = 40 mph.