Question

Give an example to show that the order of iterated limits can be interchanged although the simultaneous limit does not exist

Answer 1

Answer:

One has an expression whose value depends on at least two variables, one takes the limit as one of the two variables approaches some number, getting an expression whose value depends only on the other variable, and then one takes the limit as the other variable approaches some number. This is not defined in the same way as the limit

{\displaystyle \lim _{(x,y)\to (p,q)}f(x,y),\,} \lim_{(x,y) \to (p, q)} f(x, y), \,

which is not an iterated limit. To say that this latter limit of a function of more than one variable is equal to a particular number L means that ƒ(x, y) can be made as close to L as desired by making the point (x, y) close enough to the point (p, q). It does not involve first taking one limit and then another.