Multivariable Function Limits: UNIZOR.COM - Math4Teens - Calculus - Limi...

Notes to a video lecture on http://www.unizor.com

Multivariable Limits

Sometimes we have to deal with functions of two or more arguments and have to analyze the behavior of such functions as their arguments approach certain values.

The usual way to analyze this situation is to fix all arguments except one and see what happens with the function if that single argument approach the value we are interested in.
The result of this process is the reduction of variables by one, and we can repeat the same thing for the next argument, then the next etc.

For example, consider a function
F(x,y,z) =
= arctan(x) + 2^−y + z/(z+1)
and its behavior when all arguments increase without restriction.
1. Fix x and y, let z→+∞.
lim_z→+∞ z/(z+1) = 1
2. Our function now can be written as
F(x,y,+∞) = arctan(x)+2 ^−y+1
Fix x, let y→+∞.
lim_y→+∞ 2^−y = 0
3. Our function now is
F(x,+∞,+∞) = arctan(x)+0+1
Let x→+∞.
lim_x→+∞ arctan(x) = π/2
The limit of our function when z→+∞, y→+∞, x→+∞ is
lim F(x,y,z) = π/2 + 1

This is great, but we have a result that seems to depend on the order of arguments we analyze.
Can the result be different if we choose a different order, say, fix z and y getting a limit by x, then fix z getting a limit by y and finish with a limit by z?

The answer is:
Under certain relatively broad conditions, the final result is not dependent on the order of taking limits by different arguments.

We consider only a case of a function of two arguments (in a general case of many arguments the proof is similar but a bit more tedious) and prove the theorem below.

Prior to that, let's talk about a particular type of taking a multivariable function to a limit - a uniform convergence.
Assume, we have to analyze the limit of function F(x,y) as x→a and y→b.
We say that function F(x,y) is converging uniformly by y to Y(y) as x→a if for any positive ε there is positive δ such that
|F(x,y)−Y(y)| < ε
as long as both |x−a| < δ and |y−b| < δ.

In other words, as x→a, not only F(x,y)→Y(y) for any specific value of y (so-called point convergence of F(x,y) to Y(y)), but F(x,y) gets equally close to Y(y) (distance less than small ε) for all values of y within certain neighborhood of its limit point y=b.

Theorem

Assume that all limits below exists and satisfy the conditions listed.
IF
lim_y→b F(x,y) = X(x)
         uniformly by x
lim_x→a X(x) = A
lim_x→a F(x,y) = Y(y)
         uniformly by y
lim_y→b Y(y) = B
THEN
A = B

Proof

Assume, A ≠ B.
Let |A−B|=d.
Let ε=d/4.

Then the following logic holds.

lim_x→a F(x,y) = Y(y) (uniformly by y) ⇒
⇒ ∃δ_a: |Y(y)−F(x,y)| < ε ∀x∈[a−δ_a,a+δ_a] and
∀y∈[b−δ_a,b+δ_a]

lim_y→b Y(y) = B ⇒
⇒ ∃δ_b: |B−Y(y)| < ε
∀y∈[b−δ_b,b+δ_b]

Let δ = min(δ_a,δ_b)
Then from both above statements follows that within a small neighborhood of limit points x=a and y=b
∀x∈[a−δ,a+δ] and
∀y∈[b−δ,b+δ]
the following is true
|B−Y(y)|<ε and
|Y(y)−F(x,y)|<ε
from which, in turn, follows
|B−F(x,y)|<2ε=d/2

That is, within a small distance of arguments x and y around their limit points a and b the value of F(x,y) deviates from B by less than d/2.

Now repeat the same from the position of another limit of function F(x,y).

lim_y→b F(x,y) = X(x) (uniformly by x) ⇒
⇒ ∃γ_b: |X(x)−F(x,y)| < ε ∀x∈[a−γ_b,a+γ_b] and
∀y∈[b−γ_b,b+γ_b]

lim_x→a X(x) = A ⇒
⇒ ∃γ_a: |A−X(x)| < ε
∀x∈[a−γ_a,a+γ_a]

Let γ = min(γ_a,γ_b)
Then from both above statements follows that within a small neighborhood of limit points x=a and y=b
∀x∈[a−γ,a+γ] and
∀y∈[b−γ,b+γ]
the following is true
|A−X(x)|<ε and
|X(x)−F(x,y)|<ε
from which, in turn, follows
|A−F(x,y)|<2ε=d/2

That is, within a small distance of arguments x and y around their limit points a and b the value of F(x,y) deviates from A by less than d/2.

Make our neighborhood even smaller by choosing β=min(γ,δ) and consider
∀x∈[a−β,a+β] and
∀y∈[b−β,b+β]

Now we see that within this neighborhood F(x,y) is closer to A by less than d/2 and closer to B by less than d/2, where d is the distance between A and B, which is impossible.

Therefore, A = B.