Is it necessary to define the partial derivative of a function at a certain point?

It is defined by partial derivative
The partial derivative of function f (x, y) at (0,0) is defined as LIM (x - > 0, Y - > 0) (f (x, 0) - f (0,0)) / (x-0). If there is no definition at (0,0), then the partial derivative is meaningless

Can the partial derivative of multivariate function be defined on the boundary point, similar to unilateral limit

In fact, from the definition of multivariate function limit: in the definition of limit, it does not require the function to have a definition in the neighborhood of point P. in the process of point x → P, we only need to take the value of X in the intersection of the neighborhood of P and the function definition domain. From this definition, we can find the limit at the boundary point, since we can find the limit

The non differentiable limit of a function doesn't exist, does it
If the function is not admissible at x = XO, and the limit does not exist, right
Why is y = | x |, the absolute value of X, not differentiable at 0, but the limit is 0

For example, y = | x |, the absolute value x, is not differentiable at 0, but the limit is 0

Multivariate function proves that limit does not exist
Such as the title

Let x = y (i.e. let x = y and go to zero), and the limit value is 1
Take x ^ 2 = y, calculate the limit value as 0, unequal
So the limit doesn't hold

What are the methods to prove the nonexistence of function limit

The sufficient and necessary condition for the existence of (x - > A) function limit is that both left and right limits exist and are equal. If this condition is not satisfied, then the limit does not exist. Specifically, the left limit does not exist, the right limit does not exist, and both left and right limits exist but are not equal
(x - > A or X - > ∞) if two columns of XN can be selected so that f (xn) tends to two different limit values, then the limit does not exist

Master to see why the limit of the binary function does not exist
Find the limit of F (x, y) = XY ^ 2 / (x ^ 2 + y ^ 4) at (0,0). I calculate it all 0, but why does the answer say it doesn't exist?
ground floor:
What I did before is to set y = KX, and the result is 0. Why should I set X = KY ^ 2?

This is because the limit of binary function is calculated by considering the approach of the coordinates (x, y) of the midpoint in the rectangular coordinate system. Ha, it can approach from any direction on the plane, not from left and right directions. Here, you set X = KY ^ 2, that is to say, the point (x, y) approaches (0,0) by the trajectory of the curve X = KY ^ 2

The limit of two functions does not exist. Does the sum of them exist?
As the title

For example, when x tends to infinity, Lim X and lim (- x) do not exist, but
lim（x+（-x））=0
Lim X and lim 2x do not exist, and lim x + 2x does not exist

For multivariate function, if its limit exists, can we deduce that it is continuous at that point?

No, if the limit value at that point is equal to the function value at that point

Is the limit of bivariate function continuous? Is it multivariate? Please give an example or prove it

No, no matter one, two or more variables, limit has nothing to do with continuity;
Limit refers to: when a point is infinitely close to a fixed point, but never equal to the fixed point, the value of the function has nothing to do with whether the function is defined at this certain point;
You can think of "removable discontinuity". The limit of function exists at the point of removable discontinuity, which is obviously discontinuous

The definition of function limit
How to prove that the limit of E Λ x at x = 0 is 1 by definition

As long as we prove that the value of (e ^ x-1) is in the neighborhood of 0, we can always find an X such that (e ^ x-1) is less than δ
First of all, let's order
The limit of (e ^ x-1) = δ
Then we can calculate x = ㏑ (δ + 1)
Now let's take 0 ＜ x1 ＜㏑ (δ + 1)
Since (e ^ x-1) is an increasing function, the value of this function is less than δ when X1 is brought in
Therefore, if the value of (e ^ x-1) is any value δ in the neighborhood of 0, δ > 0, we can always find an X such that (e ^ x-1) is less than δ

Is it necessary to define the partial derivative of a function at a certain point?