First of all, the answer depends on the definitions. If you allow +infinity = infinity and -infinity for the value of a multidimensional limit, then the answer is infinity. Here are the arguments. The numerator approaches 1 and the denominator approaches 0 , being nonnegative, if x->1, y->0 as follows from the taylor expansions

mtaylor((1+y)^(x-1), [x = 1, y = 0], 5);

mtaylor(1-cos((x-1)^2+y^2)^(1/4), [x = 1, y = 0], 5);

.

Therefore, the limit

limit((1+y)^(x-1)/(1-cos((x-1)^2+y^2)^(1/4)),{x=1,y=0});

is +infinity.

The point (1,0) is a limit point of the domain of the function under consideration so the twodimensional limit is correctly defined.

Also if the limit (under consideration) exists, the one is equal to the both iterated limits

limit(limit((1+y)^(x-1)/(1-cos((x-1)^2+y^2)^(1/4)), x = 1), y = 0);

infinity

and

limit(limit((1+y)^(x-1)/(1-cos((x-1)^2+y^2)^(1/4)), y = 0), x = 1);

infinity

If the infinite limits are not allowed, then the limit does not exist.