102 CHAPTER 6. MULTI-VARIABLE CALCULUS

and if g is scalar valued with limy→x g(y) = K ̸= 0,

limy→x

f(y)g(y) = LK. (6.5.3)

Also, if h is a continuous function defined near L, then

limy→x

h◦ f(y) = h(L) . (6.5.4)

Suppose limy→x f(y) = L. If |f(y)−b| ≤ r for all y sufficiently close to x, then |L−b| ≤ ralso.

Proof: The proof of 6.5.1 is left for you. It is like a corresponding theorem for con-tinuous functions. Now 6.5.2is to be verified. Let ε > 0 be given. Then by the triangleinequality,

|f ·g(y)−L ·K| ≤ |fg(y)− f(y) ·K|+ |f(y) ·K−L ·K|≤ |f(y)| |g(y)−K|+ |K| |f(y)−L| .

There exists δ 1 such that if 0 < |y−x|< δ 1 and y ∈ D(f) , then

|f(y)−L|< 1,

and so for such y, the triangle inequality implies, |f(y)| < 1 + |L| . Therefore, for 0 <|y−x|< δ 1,

|f ·g(y)−L ·K| ≤ (1+ |K|+ |L|) [|g(y)−K|+ |f(y)−L|] . (6.5.5)

Now let 0 < δ 2 be such that if y ∈ D(f) and 0 < |x−y|< δ 2,

|f(y)−L|< ε

2(1+ |K|+ |L|), |g(y)−K|< ε

2(1+ |K|+ |L|).

Then letting 0 < δ ≤min(δ 1,δ 2) , it follows from 6.5.5 that

|f ·g(y)−L ·K|< ε

and this proves 6.5.2.The proof of 6.5.3 is left to you.Consider 6.5.4. Since h is continuous near L, it follows that for ε > 0 given, there exists

η > 0 such that if |y−L|< η , then

|h(y)−h(L)|< ε

Now since limy→x f(y) = L, there exists δ > 0 such that if 0 < |y−x|< δ , then

|f(y)−L|< η .

Therefore, if 0 < |y−x|< δ ,

|h(f(y))−h(L)|< ε.