Multivariable Function Limits: UNIZOR.COM - Math4Teens - Calculus - Limit of Function

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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.