How Do You Solve This Multivariable Limit Problem?

[Hessam]

ok... so I'm taking calc3, and didnt go to class for a month, lol

now i can't solve this (what seems like) a simple limit... please help


lim
(x,y,z) -> (0,0,0)

of this

xy + yz + zx
-----------------
(x^2)+(y^2)+(z^2)




i tried doing the limit as (x,y,z) approaches (x,0,0) (0,y,0) and (0,0,z) and got zero, but that's not the answer

however when i put it into my calc and grind it out i get 1... but how can i do it on paper?

thankyou very much in advance

 

[Maxos]

The limit does not exist

To show that, take two curves, e.g:

$$x=y=z$$ and $$x=y=-z$$

You'll find $$\frac {3x^2} {3x^2}$$ and $$\frac {-x^2} {3x^2}$$

Whose limits are 1 and $$-\frac {1} {3}$$, which are different. CVD

 

[thepatientmental]

First of all, don't worry about not attending class for a month. It happens to all of us sometimes and it's never too late to catch up. Now, let's tackle this limit problem.

To solve this limit, we can use the squeeze theorem. This theorem states that if we have two functions, f(x) and g(x), such that f(x) ≤ h(x) ≤ g(x) for all x near a, and if lim f(x) = lim g(x) = L, then lim h(x) = L.

In this case, we can rewrite the given function as:

(x^2 + y^2 + z^2)/3 * (xy + yz + zx)/ (x^2 + y^2 + z^2)

Now, we can see that the first part of the function, (x^2 + y^2 + z^2)/3, approaches 0 as (x,y,z) approaches (0,0,0). Therefore, we can use the squeeze theorem and say that the limit of the entire function is equal to the limit of the second part, (xy + yz + zx)/ (x^2 + y^2 + z^2).

To find the limit of this second part, we can use the method of plugging in values. Let's try approaching the limit from the x-axis, i.e. when y and z are both 0. In this case, the function becomes:

lim (x,0,0) -> (0,0,0) of (x*0 + 0*0 + 0*x)/ (x^2 + 0^2 + 0^2)

= lim (x,0,0) -> (0,0,0) of 0/ x^2

= lim (x,0,0) -> (0,0,0) of 0

= 0

Similarly, if we approach the limit from the y-axis or the z-axis, we get the same result of 0. Therefore, we can conclude that the limit of the second part is also 0.

Using the squeeze theorem, we can say that the limit of the entire function is also 0. Therefore, the final answer is 0.

I hope this helps and don't hesitate to ask for clarification if needed. Keep practicing and you'll become more comfortable with limits. Good luck!