M408D

Fun Problems

• Sequences and series
  1. evaluate sqrt(2 + sqrt(2 + sqrt(2 + ...)))
• L'Hopital's Rule
  1. l'hopital's rule is for evaluating limits of the form 0/0 or infinity/infinity. it says that if lim [x->a] f(x)/g(x) has one of these forms, you can replace f(x)/g(x) with f'(x)/g'(x). with this in mind, what is wrong with this proof that 1 = -1?
     1 = lim [x->inf] (x + sin x²)/(x - sin x²)
     = lim (1 + 2x cos x²)/(1 - 2x cos x²) (here we used l'hopital)
     = lim (1/x + 2 cos x²)/(1/x - 2 cos x²) (divide top and bottom by x)
     = lim (2 cos x²)/(-2 cos x²) (here we replace 1/x with 0 since x -> infinity)
     = lim 2/-2
     = -1
     the first equality is true because the sine function is bounded and so x +/- sin(anything) is dominated by x as x->infinity.