Suppose a fish swimming a distance of L ft at a speed of v fit/sec relative to the water against a current flowing at the rate of u ft/sec (u<v) expends a total energy given by

E(v)= (aLv^3)/(v-u)
Where E is measured in foot-pounds and a is a constant.

a) Evaluate the limit of E(v) as v-->u+ and interpret your result
b) Evaluate the limit of E(v) as v-->infinity and interpret your result

Could you please show me the math behind answer a? How did you calculate the limit to get infinity?

a) as v -> u the Energy required becomes infinite because the fish does not move and requires infinite time to go a distance L.

b) as v--> infinity, the energy required is proportional to v^2 (which is true for the fluid drag force), as the stream velocity becomes negligible compared to v.

Energy = force x distance.

To evaluate the limits and interpret the results, we'll substitute the values into the given expression for E(v) and analyze the behavior.

a) Evaluating the limit of E(v) as v approaches u from the right (v → u+):

Let's substitute v = u+h, where h approaches 0 as v approaches u.

E(v) = (aL(u+h)^3) / ((u+h) - u)
= (aL(u+h)^3) / h

Now, we can simplify this expression:

E(v) = aL(u^3 + 3u^2h + 3uh^2 + h^3) / h
= aL(u^3/h + 3u^2 + 3uh + h^2)

Taking the limit as h approaches 0:

lim (h → 0) of E(v) = aL(u^3/0 + 3u^2 + 3uh + 0^2)
= infinity

Interpretation: The limit of E(v) as v approaches u from the right is infinite. This means that as the fish's velocity approaches the current velocity from below, the amount of energy expended by the fish becomes infinite. In other words, the fish needs an infinite amount of energy to swim against the current at the same speed as the current.

b) Evaluating the limit of E(v) as v approaches infinity:

Let's substitute a large value for v, assuming u is a constant, and evaluate the limit.

E(v) = (aLv^3) / (v-u)
= (aL(v^3)) / v
= aLv^2

Taking the limit as v approaches infinity:

lim (v → ∞) of E(v) = aL∞^2
= infinity

Interpretation: The limit of E(v) as v approaches infinity is infinite. This means that as the fish's speed relative to the water increases without bounds, the amount of energy expended by the fish also becomes infinite. In other words, the fish needs an infinite amount of energy to maintain a high speed against the current.

In summary, as the fish's velocity approaches the current velocity from below, or as the fish's speed relative to the water increases without bounds, the amount of energy it takes for the fish to swim against the current becomes infinitely large.