Question

Use a computer algebra system to draw a direction field for the differential equation y' = 3y^3 − 12y. Get a printout and sketch on it solutions that satisfy the initial condition y(0) = c for various values of c.

A)For what values of c does the limit L = lim t → [infinity] y(t) exist?

a.c is in [−2, 2]
b.c is in [2, [infinity])
c.c is in [−2, 0]
d.c is in [0, 2]
e.c is in (−[infinity], −2]

B) What are the possible values for this limit L? (Enter your answers as a comma-separated list.) L =

Answer 1

Answer: simplest reason is that Infinity is not a number, it is an idea.

So 1∞ is a bit like saying 1beauty or 1tall.

Maybe we could say that 1∞= 0, ... but that is a problem too, because if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?

In fact 1∞ is known to be undefined.

But We Can Approach It!

So instead of trying to work it out for infinity (because we can't get a sensible answer), let's try larger and larger values of x:

graph 1/x

x 1x

1 1.00000

2 0.50000

4 0.25000

10 0.10000

100 0.01000

1,000 0.00100

10,000 0.00010

Now we can see that as x gets larger, 1x tends towards 0

We are now faced with an interesting situation:

Explanation: