- #36
Inventive
- 85
- 12
See my note
Stephen Tashi said:That's correct, but does that, by itself, prevent the limit from existing? This is very technical question involving how we interpret the definition of the limit. The definition has the form " For each ... there exists...such that...for each ##(x,y)##, ##| f(x,y) - L| < \epsilon##". ##\ ## Are we to understand that the implicit meaning is "for each ##(x,y)## in the domain of ##f##"? ##\ ## Or perhaps that ##| f(x,y) - L | < \epsilon##" is not a false statement when ##(x,y)## is not in the domain of ##f##, but rather an undefined expression?
That's a good idea, but does it work? ##\ | sin(\theta) | \le |\theta | ##. ##\ \theta## would be a polynomial of degree 12 in ##y## and the denominator would be a polynomial of degree 4 in ##y##.
Thanks for taking the time to explain it this way. I can visualize what you are sayingCitan Uzuki said:All very true, but actually orthogonal to the point I was trying to make. What I was trying to get at is that if you are trying to find the limit of a rational function (or something which is effectively a rational function, since [itex]\sin (\theta) ~ \theta[/itex] for small [itex]\theta[/itex]) at the origin, the denominator being zero along some curve running through the origin usually means that you can make the function blow up by following another curve close to the first one. Thus seeing that the function fails to exist along some curve through the origin is usually a good sign that the limit won't exist, independently of the answer to that technical question.
Yes. Along the curve x=-y^2 + y^4:
[tex]\lim_{y \to 0^+} \frac{\sin (x^3 + y^3)}{x + y^2} \\
= \lim_{y \to 0^+}\frac{\sin ((-y^2 + y^4)^3 + y^3)}{-y^2 + y^4 + y^2} \\
= \lim_{y \to 0^+}\frac{\sin (y^3 - y^6 + 3y^8 - 3y^{10} + y^{12})}{y^4} \\
= \lim_{y \to 0^+}\frac{y^3 + O(y^6)}{y^4} \\
= \lim_{y \to 0^+}\frac{1 + O(y^3)}{y} \\
= \infty[/tex]
So the original limit does not exist.
Citan Uzuki said:Yes. Along the curve x=-y^2 + y^4:
You're right. The first example of your link is even used in stewart's textbook. Thanks for noting!Stephen Tashi said:That is not a valid conclusion. The existence of a common value for the limit along a family of straight line paths does not imply that the limit as (x,y) -> (0,0) exists. For example, see the answer to http://math.stackexchange.com/quest...existence-of-limit-of-a-two-variable-function
The limit of a two variable function is the value that the function approaches as the two variables approach a particular point or value. It is also known as the "limit point" or "limiting value".
The limit of a two variable function is calculated by evaluating the function at different points that approach the given point, and observing the trend of the values. If the values of the function approach a single value as the points get closer to the given point, then that value is considered as the limit.
The limit of a two variable function helps us understand the behavior of the function at a particular point. It also helps in determining the continuity of the function at that point, and in finding the derivatives of the function.
No, the limit of a two variable function can only have one value. This value may be the same or different from the value of the function at the given point.
The limit of a two variable function may be different for different directions of approach. This is known as the directional limit. It is important to consider the direction of approach when calculating the limit of a two variable function.