I have to find the limit x-->0 arctan x cos(1/x), and justify my method, by using the squezzing princple. Can somebody help also I have another question(not as important because out teacher gave to us to think over the weekend but I have no idead where to start)
Letf(x) = x^3 + bx^2 + cx + d be a general cubic polynomial with the
coefficient in front of x3 adjusted to be a 1.
(a) Explain why f(x) > 0 when x > 0 and very large, and why f(x) < 0
whenx < 0 and very large.
Hint. Rewrite f as f(x) = x^3(1+b=x+c=x2+d=x2) and then inspect
the sign of the part in brackets when x is very large.
(b) Use the above information to show that every cubic must cut the x-axis
in at least one place, i.e. prove every cubic has a real root.
(c) Does every degree 4 polynomial cut the x-axis? Explain your answer.