Confirm Limit Existence for Function f w/o Piecewise Def.

[Haku]

Homework Statement

Defining the existence of a limit

Relevant Equations

Limits

If you are told something holds if the limit exists, and given a function f (specifically not piecewise defined), is it enough to show that the limit as x approaches c = the function evaluated at c?
With a piecewise defined function, it is easy to check both sides of a potential discontinuity, by using each part of the defined function, i.e. from either side of the potential discontinuity.
For example:

In this definition, the j-th partial derivative of f at x0 exists if that limit exists.
Does it exist if it is equal to something not infinity? Or would I need to be more rigorous and show the limits from either side exist and are all equal to each other?
An application of this definition is used in the following example:

Where I checked that all partial derivatives exist via the above definition, but when checking, for example, j-th partial derivatives at (x, y) = (0, 0) for j = 1, you just use that definition above and get that the limit = 0. And by my lecturer in class, that was sufficient.

But, was that enough? Because since we are using limits (and in analysis), rather than just assuming it exists and plugging in for specific values, wouldn't you have to actually prove that the limit exists first? Rather than just show that the limit as t approaches 0 of f(0, 0) = function evaluated at (0, 0) = 0?

Thanks.

 

[Office_Shredder]

If the limit exists, then necessarily the limit from either side exists. This is a pretty simple proof, that you should try to show yourself:

If
$$\lim_{x\to a} f(x) = L$$
Then
$$\lim_{x\to a^+} f(x) = L$$
So there's no need to check the limit from a specific direction once you know the limit exists. I think you are probably getting a bit confused because for a piecewise defined function, it's often taken as "obvious" that the limit from each direction exists because the pieces of the function are continuous functions, so the only thing you need to check is that the two limits are equal to each other. In other words, another lemma you can prove is
If
$$\lim_{x\to a^+} f(x) = L = \lim_{x\to a^-} f(x)$$
Then
$$\lim_{x\to a} f(x) = L$$I'm not sure I totally follow what you are confused about in the last part. You never evaluate the limit as t approaches 0 of f(0,0). Did you mean something like f(t,0)?

 

[Haku]

If the limit exists, then necessarily the limit from either side exists. This is a pretty simple proof, that you should try to show yourself:

If
$$\lim_{x\to a} f(x) = L$$
Then
$$\lim_{x\to a^+} f(x) = L$$
So there's no need to check the limit from a specific direction once you know the limit exists. I think you are probably getting a bit confused because for a piecewise defined function, it's often taken as "obvious" that the limit from each direction exists because the pieces of the function are continuous functions, so the only thing you need to check is that the two limits are equal to each other. In other words, another lemma you can prove is
If
$$\lim_{x\to a^+} f(x) = L = \lim_{x\to a^-} f(x)$$
Then
$$\lim_{x\to a} f(x) = L$$I'm not sure I totally follow what you are confused about in the last part. You never evaluate the limit as t approaches 0 of f(0,0). Did you mean something like f(t,0)?

My initial question was not really meant to ask about "If the limit exists, then necessarily the limit from either side exists" as this is straightforward. Rather it is this: We don't know the limit exists, yet we act as if it has (in some cases), but (from an analysis standpoint) we should always prove that it does exist right?
In terms of my example of the existence of partial derivatives, showing that limit exists means showing the limit from either side exists and are equal? But what about higher dimensions? i.e. greater than 2 dimensions? How would you go about this?

I think I was confused about the piecewise defined example too. Because I didn't know with piecewise functions you suppose all parts are continuous, is this standard assumption when given a piecewise defined function?

Thanks!

 

[vela]

If you are told something holds if the limit exists, and given a function f (specifically not piecewise defined), is it enough to show that the limit as x approaches c = the function evaluated at c?

That would be showing $f$ is continuous at $x=c$, not that the limit exists.

Where I checked that all partial derivatives exist via the above definition, but when checking, for example, j-th partial derivatives at (x, y) = (0, 0) for j = 1, you just use that definition above and get that the limit = 0. And by my lecturer in class, that was sufficient.

But, was that enough? Because since we are using limits (and in analysis), rather than just assuming it exists and plugging in for specific values, wouldn't you have to actually prove that the limit exists first?

If you can calculate the limit, you know it exists. You just have to be careful that you're calculating the limit correctly.

Rather than just show that the limit as t approaches 0 of f(0, 0) = function evaluated at (0, 0) = 0?

Again, this is continuity, not showing that the limit exists.

 

[vela]

My initial question was not really meant to ask about "If the limit exists, then necessarily the limit from either side exists" as this is straightforward. Rather it is this: We don't know the limit exists, yet we act as if it has (in some cases), but (from an analysis standpoint) we should always prove that it does exist right?

I think you're making this unnecessarily complicated.

Consider just plain old function $f: \mathbb{R} \to \mathbb{R}$. We say $f$ is differentiable at $x=c$ if the limit
$$\lim_{h \to 0} \frac{f(c+h)-f(c)}{h}$$ exists. But if you were to ask is the function $f(x) = x^2$ differentiable at $x=1$, you don't worry about proving that the limit exists. You just differentiate $f$ and see that nothing weird is happening at $x=1$. The rule you learned about how to differentiate $x^n$ guarantees the limit exists.

In terms of my example of the existence of partial derivatives, showing that limit exists means showing the limit from either side exists and are equal? But what about higher dimensions? i.e. greater than 2 dimensions? How would you go about this?

In one dimension, for a limit to exist, it has to be true that the one-sided limits from both sides are equal. To word it more intuitively, that means it shouldn't matter from which direction you approach a point. The same idea applies to higher dimensions. It's just not "from either side" anymore, but more generally, it's from any direction.

In the case of the partial derivatives, however, you're picking out one particular direction, so it's really a one-dimensional limit, not a limit in higher dimensions.