Find the limit as x approaches infinity?

Question

this is the function: x^(2)log(x)sin(1/x)

Answer 1

Using Excel I generated the following table:

10 100 1 0.001749999 0.174999911

100 10000 2 0.000175 3.499999982

1000 1000000 3 1.75E-05 52.5

10000 100000000 4 1.75E-06 700

100000 10000000000 5 1.75E-07 8750

1000000 1E+12 6 1.75E-08 105000

10000000 1E+14 7 1.75E-09 1225000

100000000 1E+16 8 1.75E-10 14000000

1000000000 1E+18 9 1.75E-11 157500000

10000000000 1E+20 10 1.75E-12 17500000...



The left hand column is x, followed by x^2, followed by log(x), followed by sin(1/x), and finally followed by [x^2 * log(x) * sin(1/x)] where * is the multiplication symbol.

As you can see, as x --> infinity, so does your function.  

If you have had an advanced course in Limit Theory, you would realize that while the log(x) is insignificant, the x^2 * sin(1/x) is dominated by the x^2, while x * sin(1/x) is not.  This requires delta - epsilon proofs.  I chose to guess that you would like to see the product of Excel , followed by the basic concepts.  I hope that this is what you wanted.