Question

check my Q1 answer and do Q6 thanks!

plz check my answer in Q1 and do Q6 thanks!

Problems For ach peoldem other tane y will reeive a mark ef 1/s if yos do not awer. (But you esd the mon-anewer to get credit for it nly. No peoof is required for this probidem, mp sae the inly which mak the (5 points) Peove the lowing fact, which Yong'st s 1 Before starting the problems Define f R2 (0,0)R by where αι, α2, β1,β2 are assumed to be positive Consider lim(r, mz*) for different choices of m, 6, and ask yourself: under what assumptions on a1, o2,1, 2 do you think that you can prove that the limit does not erist? In other words along what paths of the form γ(x) (z.ma.δ) does f not have a limit? If we are optimistic, we can hope that the limit exists whenever you cannot show that it does not erist. Based on these thought experiments, yo u can formulate a conjecture that looks like this: My Conjecture 1. The limit lim(z,y)→(0,0) f(x,y) exists if and only if (2) α.Gfdf Ra( S- Pa It is up to you to find the correct necessary and sufficient condition in (2). In the homework assignment, you will prove that the conjecture, as you have formulated it, is , β2). inequality involving 01,02.31 indeed true 1. (1 point) Complete the statement of Conjecture 1 by finding inequality (2) explic itly. No proof is required for this problem, simply state the inequality which makes the conjecture true. 6.) (5 points). Prove the other half of the conjecture, by showing that lim(z.y)(0,0) f(x,y) exists if the condition that you found is satisfied. What is the limit, when it exists? For the benefit of the person who wi mark the question, please state clearly what you are proving. Hints: 1. Try to write f as a product f = f1/2 for functions fl and f2 such that fl

Answer 1

Your answer to question 1 seems incorrect as you have fixed delta which must be kept arbitrary to prove continuity.

Alone γ (x): (x, mx +116 alona Y Noh lim f(x ) exists 。 pende equadlity halas Conjecture The limit lim fet) exists