In this problem we show that the function f(x, y) 7x²-y² lim (2,2x)+(0,0) z²+y² 7x² - y² x² + y² does not have a limit as (x, y) → (0,0). (a) Suppose that we consider (x, y) → (0,0) along the curve y = 2x. Find the limit in this case: (b) Now consider (x, y) → (0,0) along the curve y = 3x. Find the limit in this case: 7x²-y² lim (2,3x)+(0,0) z²+y² (c) Note that the results from (a) and (b) indicate that ƒ has no limit as (x, y) → (0,0) (be sure you can explain why!). To show this more generally, consider (x, y) → (0, 0) along the curve y = mx, for arbitrary m. Find the limit in this case: lim (z,mx)+(0,0) z²+y² 72²-y² = (Be sure that you can explain how this result also indicates that ƒ has no limit as (x, y) → (0,0).

Given: x(x+ 8)(x – 2) (x +2)³ x3 – 4x2 8(7x – 4) (x + 2)4 f(x) = with f'(x) = and f"(x) (x + 2)2 ' a. Find the domain of f and the intercepts of the graph of f. b. Using limits, show that x = -2 and y = x – 8 are asymptotes of the graph of f. c. Identify the critical numbers of f and the possible points of inflection of the graph of f. d. Construct a table of signs for f' and f". From here, specify the following: i. the intervals on which the graph of f is increasing, decreasing, concave up, and concave down, and ii. the values of x at which ƒ has relative extrema, inflection points, and vertical asymp- totes (if any). e. Sketch the graph of f with emphasis on concavity. Label all the intercept points, relative extremum points, and inflection points with their coordinates. Show the asymptotes of the graph of f and label them with their respective equations.

Vx - Vy+9 lim = (х, у)--(11,2) х+у+9 х — у — 9

On which interval is the function f(x) = x2 - 2x - 8 decreasing?  A [-2, 4]  B (-∞, 4)  C (-2, 4)  D (-∞, 1)  E (1, ∞)

For the function f(x) = 4x + 19x + 28, find f'(x) using the f(x +h)- f(x) definition lim O 4r 8x O 8r O 38x +28 O 8&r + 19 O 4c + 19

Q (1): Find fxyz for the function: a) f(x,y, z) = xe? + ye* + zey

Let f(x) = Vx +7 and g(x) = x +3 a) Find (2) (2) show work. b) Give the domain of the function (-) x) in interval notation