Piecewise Functions, Derivatives and Limits are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Please help with the following problems regarding limitsandpiecewise functions.
1) Let f(x) {o if x is a natural odd number
{1 otherwise
Does f(x) have a limit as x approaches infinity? Explain you answer.
2) Let f(x) {1 if x is a natural odd number
{ 1-1/x

Determine whether or not each of the following limits exists. Discuss also the continuity of each of the following functions at given point c. Give reasons to your answers.
Please see the attached file for the fully formatted problems.

1. Use logarithmic differentiation to find [see attachment].
2. Use logarithmic differentiation to find [see attachment].
3. Use logarithmic differentiation to find [see attachment].
Please see attachment.

Find the derivatives of the following functions:
(i) f(x) = sqrt(x)*(2x^3-4) + 3x^(-1/4)
(ii) y(x) = (x(x^2-1))/(x^3-4)
(iii) g(u) = (4u^(1/3))*(sqrt(u^3+1))
Please see attached and show step by step, thanks.

I need some help with these three questions:
A. Evaluate lim x->2 for (5x^2 + 60x + 100)/ (x^3 + 4x^2 + 4x)
B. Evaluate the integral (256,1) for ((x^1/2)/4 + 8/x^8)) dx
C. Evaluate the integral (6,2) for (10x-72x^-3) dx
See attachment for better formula representation

1. Find the derivatives for the following functions ("^" means "to the power of", sorry I can't do double exponents on my keyboard) :
a. f(X) = 100e10X
b. f(X) = e(10X-5)
c. f(X) = e^X3
d. f(X) = 2X2e^(1- X2)
e. f(X) = 5Xe(12- 2X)
f. f(X) = 100e^(X3 + X4)
g. f(X) = e^(200X - X2 + X100)
2. Fi

A car is traveling on a straight road with velocity 55 ft/sec at time t = 0. For 0 ≤ t ≤ 18 seconds, the car's acceleration a(t) , in ft/sec2, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at t = 2 seconds? Why or why not?
(b) At what time in the interval 0 &

Please see attachment.
4.10 Find the partial derivatives with respect to x, y, and z of the following functions:
(a) f(x,y,z) = ax^2 +bxy + cy^2
(b) g(x,y,z) = sin(axyz^2),
(c) h(x,y, z) = ax^(xy/z^2).
where a, b, and c are constants