228 CHAPTER 12. VECTOR VALUED FUNCTIONS

10. Let

f (x,y)≡

{2x2−y2

x2+y2 if (x,y) ̸= (0,0)

0 if (x,y) = (0,0).

Find lim(x,y)→(0,0) f (x,y) if it exists. If it does not exist, tell why it does not exist.Hint: Consider along the line y = x and along the line y = 0.

11. Find the following limits if possible

(a) lim(x,y)→(0,0)x2−y2

x2+y2 .

(b) lim(x,y)→(0,0)x(x2−y2)(x2+y2)

= 0.

(c) lim(x,y)→(0,0)(x2−y4)

2

(x2+y4)2 . Hint: Consider along y = 0 and along x = y2.

(d) lim(x,y)→(0,0) xsin(

1x2+y2

).

(e) lim(x,y)→(1,2)−2yx2+8yx+34y+3y3−18y2+6x2−13x−20−xy2−x3

−y2+4y−5−x2+2x . Hint: It might help towrite this in terms of the variables (s, t) = (x−1,y−2) .

12. Suppose limx→0 f (x,0) = 0 = limy→0 f (0,y). Does it follow that

lim(x,y)→(0,0)

f (x,y) = 0?

Prove or give counter example.

13. f : D⊆ Rp→ Rq is Lipschitz continuous or just Lipschitz for short if there exists aconstant K such that

|f (x)−f (y)| ≤ K |x−y|

for all x,y ∈D. Show every Lipschitz function is uniformly continuous which meansthat given ε > 0 there exists δ > 0 independent of x such that if |x−y| < δ , then|f (x)−f (y)|< ε .

14. If f is uniformly continuous, does it follow that |f | is also uniformly continuous?If |f | is uniformly continuous does it follow that f is uniformly continuous? An-swer the same questions with “uniformly continuous” replaced with “continuous”.Explain why.

15. Let f be defined on the positive integers. Thus D( f ) = N. Show that f is auto-matically continuous at every point of D( f ). Is it also uniformly continuous? Whatdoes this mean about the concept of continuous functions being those which can begraphed without taking the pencil off the paper?

16. Let

f (x,y) =

(x2− y4

)2

(x2 + y4)2 if (x,y) ̸= (0,0)

Show limt→0 f (tx, ty) = 1 for any choice of (x,y). Using Problem 11c, what doesthis tell you about limits existing just because the limit along any line exists.