Evaluate the limits using lhopitals rule - limx-gt0

Question

1) Find the absolute maximum and the absolute minimum for the function f(t)= 2cost + sin2t at the interval [O, pi/2].

2) For the function f(x) = x²/(x-1), analyze and find the following:

i. domain

ii. vertical, horizontal, and slant asymptotes

iii. local maximum and local minimum

iv. intervals where the function is concave upwards, where it is concave downwards, and points of reflection

v. graph the function using the results above and any other useful properties

3) Evaluate the limits using L'Hopital's Rule

i. lim(x->0) (e^x-e^-x-2x)/(x-sinx)

ii. lim (x->infinity) (1+(a/x))^b/x

4) A box with a square base and open top must have a volume of 32,000cm³. Find the dimensions of the box that minimize the amount of material used.