Please submit the answers to the following in electronic format, either TeX, Word or PDF.

(Be as rigorous and precise as you can!)

1. Show that lim_{x → 0} sin(x)/x = 1, using the ε-Δ proof.
   [Hint]

2. Using the result of the previous problem, show that lim_{x → 0} sin(sin(x))/x exists
   [Hint]

3. Show that lim_{x → 0} sin(1/x) does not exist using the ε-Δ proof.
   [Hint]

4. Show that as x → ∞ [√(x+1)-√(x)] = 0
   [Hint]

5. Show that the Fibonacci sequence of numbers 1,1,2,3,5,8,13, given by the relationship a_n+1=a_n+a_n-1 satisfies

   [Hint]

   Given this recursion relation, find the asymptotic limit of a_n as n → ∞. This means find a c and an r such that

   lim_{n → ∞} a_n/(crⁿ) = 1

6. Show that one can compute π by the infinite series
   π = 4*(1-1/3+1/5-1/7+1/9-1/11+1/13-1/15+1/17-...)
   =∑₀ ^∞ (-1)ⁿ 1/(2n+1) that is from the sum of the alternating series of reciprocals of odd numbers.
   [Hint]

   Extra Credit; Derive as many of the series representations for Pi as you can. There are 43 of them listed in this reference!

7. Show that the non-alternating series ∑_n=1^n=∞ 1/(2n+1) diverges, that is, becomes infinite.
   [Hint]

8. In the section "Archimedes the Numerical Analyst" in the reference Archimedes There is a procedure to calculate π to 18 decimal places strictly from inscribed and circumscribed regular polygons of 3, 6, 12, 24, 48, and 96 sides respectively. Carry out this procedure, steps 1-7, or this reference and show that it is correct.

   See this reference (Section 1.3) for the correct relationship
   [Hint]

9. Define the Heaviside function as H(t) = 0 if t < 0,
   and
   H(t) = 1 if t >= 0 Show, using an ε-Δ proof that the limit of H(t) as t → 0 does not exist.
   [Hint]

10. Show that lim _{x → 0+} x^x = 1

    lim _{x → 0+} means that x approaches zero, but only takes on positive values...
    [Hint]