3b. Let f: AR, where ACR and c E A' and lime-c f (x) = L. Then there is U CR, U- open, c E U, and there is M≥ 0, such that, for all x ≤ Un A, |ƒ (x)| ≤ M. That is, if f has limit at c then f is locally bounded at c. Use the assumption limx→c f (x) = L to show that there is 6 > 0, such that for all x € A, if 0 < x- c < 6, then |ƒ (x)| < 1+ |L|. Show that, if c A, then for all x € D (c,d), |f (x)| < 1 + |L|. ● Argue that there is UCR, U - open, c € U, and there is M≥ 0, such that, for all xEUNA, f (x)| ≤ M. What should we take for U and for M? Show that if c E A, then a € D (c,d), f (x)| ≤ max {1+ |L|, |ƒ (c)|} . Argue that there is U CR, U- open, c € U, and there is M≥ 0, such that, for all xe Un A, f (x)| ≤ M. What should we take for U and for M?

4. Suppose F: R" → R? is of the form F(r) = (f(x), g(x)), TE R". (i) Show that lim F(r) = (a,b) if and only if lim f(r) = a and lim g(x) = b. (ii) Show that F is continuous at ro if and only if f and g are both continuous at 1o-

Prove the following. a) Use the formal definition of a limit to prove that lim x+ y = a+b. (x,y)(a b) b) If z= f(x, y) =- x" where a and b are real numbers. Show that dz a(dx) b(dy) y

4. Find the following limits or determine that they do not exist: 1+1/x a. lim,0 1+1/x? 1+1/x2 b. lim,→0 1+ 1/x 1+1/(x – 1) 2+1/(x – 1)2 c. lim→1