Taylor polynomials...help?

Question

10 points to best:

1) Use Taylor Polynomials with remainder term to evaluate the following limits:

lim [log(1-x) + xe^(x/2)]/ x^3

x-->0

All of that's divided by x^3

-------------------

Find linear and quadratic Taylor poly. approximations to f(x) = x^(1/3) about the point a = 8. Bound the error in each approximation on the interval 8 ≤ x ≤ 8 + δ with δ > 0. Obtain a numerical bound on the interval [8, 8.1]

-------

The second one is the most important! Show all steps plz, thanks!

Answer 1

1. you have [ -x - x²/2 - x^3 / 3 - O(x^4) + x + x² / 2 + x^3 / 8 + O(x^4)] / x^3 and as x --->0  this is -1/3 + 1/8 = -5 /24...note O(h) means each term has an h in it.

#2. This is crank and grind and you can do it f(8) + f '(8) [x-8] + f ''(8) [x-8]² / 2  { drop the last term for linear}...for the linear the bound is max| f ' ' | [interval length]² / 2 while quadratic is   max| f ' ' ' | [interval length]^3 / 6