A problem of finding the limit of ternary function Lim(x,y,z)to(0,0,0)(xy+yz^2+xz^2)/(x^2+y^2+z^4)=? To specific steps, thank you!

A problem of finding the limit of ternary function Lim(x,y,z)to(0,0,0)(xy+yz^2+xz^2)/(x^2+y^2+z^4)=? To specific steps, thank you!

There is no reason as follows: let z = 0 (i.e. trend from xoy plane) LIM (x, y, z) to (0,0,0) (XY + YZ ^ 2 + XZ ^ 2) / (x ^ 2 + y ^ 2 + Z ^ 4) = LIM (x, y) to (0,0) XY / (x ^ 2 + y ^ 2) (I) make x = y → 0, x = 1 / 2, y → 0 respectively; (i) If the limit exists, it must be unique, that is, l

A problem of finding the limit of a function Find the limits of the following functions: lim[(x^2+x)^1/2-(x^2-x)^1/2]=? (x→∞）

(x^2+x)^(1/2)-(x^2-x)^(1/2)
=[(x ²+ x)-(x ²- x)]/[(x^2+x)^(1/2)+(x^2-x)^(1/2)]
=2x/[(x^2+x)^(1/2)+(x^2-x)^(1/2)]
=2/[(1+1/x)^(1/2)+(1-1/x)^(1/2)]
lim{(x^2+x)^(1/2)+(x^2-x)^(1/2)}
=lim 2/[(1+1/x)^(1/2)+(1-1/x)^(1/2)]
=2/(1+1)
=1

The relationship between function limit and function continuity Relationship between function limit and function continuity

The continuity of a function at a certain point means that three conditions are satisfied
1. The function is defined at this point
2. The limit of the function exists at this point
3. The function limit is equal to the function value
So we know that if the function is continuous at x0, the limit must exist at x0
On the contrary, if the limit of the function exists at x0, the function may not be continuous at x0
For example, f (x) = (x) ²- 1)÷(x-1)
It can be seen that the function f (x) is not defined at x = 1, so it is discontinuous at x = 1,
But Lim [x → 1] f (x) = 2, that is, the limit of the function exists at x = 1!

Seeking the limit of function The function f (x) = (1-2 ^ x) / 4 ^ x is simplified to f (x) = 1 / (4 ^ x) - 1 / (2 ^ x) Find f (x) when x approaches positive infinity and negative infinity My solution is to directly make x close to positive infinity, f (x) becomes negative infinity / positive infinity. Apply lobida's law to derive the molecular denominator respectively, and get LIM (x approaches positive infinity) (- 2 ^ x * LN2) / (4 ^ x * ln4) Reduced to LIM (x approaches positive infinity) (- 1) / (2 ^ x * 2), the result is 0, which is also consistent with the result of the geometric sketchpad But when LIM (x approaches negative infinity), the result is that I calculate it as negative infinity, but it should be positive infinity What's wrong with me?

answer:
When x →∞, the calculation of the landlord is correct
When x → - ∞, the numerator becomes → 1-0 = 0 and the denominator becomes 1 / ∞ = 0, which is not an infinitive of 0 / 0 type,
Nor is it an infinitive of ∞ / ∞ type, and the Robita's law cannot be used
At this time, the result is that the numerator tends to 1, the denominator tends to 0 +, and the total result tends to + ∞
Conclusion:
1. No matter whether the result is positive infinity or negative infinity, it is a definite form and can not be used by Robita's law
2. The Robita rule can only be used in two cases where the result cannot be judged: 0 / 0, ∞ / ∞
3. As long as the result can be judged, the law of Robita cannot be used

What should we learn about function limit in college mathematics

The function limit of the university should at least learn the limit of common functions and the limit solution of combined functions, and preliminarily establish the limit idea; In the process of understanding the limit function, we should always grasp the concept of infinitesimal, which is helpful to the concept of series. The introduction of series undoubtedly makes mathematical calculation enter the armed field of engineering technology

How to find the poles of a function?

Find the derivative. When = 0, judge that the direction of side 2 is increasing on the left, decreasing on the right is the maximum, decreasing on the left and increasing on the right is the minimum. If it increases or decreases at the same time, it is not. Y = x ^ 3, x = 0 is not