two examples of limit of multivariable functions

[Elena Baby]

I tried y=mx , y=x^2 , and polar coordinates(y=rsinθ ,etc.), but it doesn't seem to be the right answer(?)So, here are my questions:

 

[Steven G]

Here are my questions. Can we please see your work so we know where you need help? Have you read our guidelines? It clearly states that we do not solve problems for students, rather we expect the student to solve their own problems with our help. So please read the guideline, follow them and post back showing the work you have done.

 

[topsquark]

I tried y=mx , y=x^2 , and polar coordinates(y=rsinθ ,etc.), but it doesn't seem to be the right answer(?)So, here are my questions:

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It's not a proof in the first case but I get consistent answers by approaching the origin using y = x and y = x^2 for each example. Here's a hint: The first limit is 0. The second is undefined.

-Dan

 

[Elena Baby]

Here are my questions. Can we please see your work so we know where you need help? Have you read our guidelines? It clearly states that we do not solve problems for students, rather we expect the student to solve their own problems with our help. So please read the guideline, follow them and post back showing the work you have done.

I'm mostly stuck at the simplifying. For example, for the second one, when y=x:



I'm not sure if I simplified this correctly from the part I added the values.
btw I'm not really used to LaTex,so sorry if It looks messy.
Same with approaching the origin with y=x^2.I'm pretty sure this is going to be undefined as most of these examples my professor gave were solved as 'undefined'.

It's not a proof in the first case but I get consistent answers by approaching the origin using y = x and y = x^2 for each example. Here's a hint: The first limit is 0. The second is undefined.

-Dan

@topsquark also said it was undefined.Is there a simplifying rule I ought to know?

 

[Elena Baby]

It's not a proof in the first case but I get consistent answers by approaching the origin using y = x and y = x^2 for each example. Here's a hint: The first limit is 0. The second is undefined.

-Dan

How do you simplify lim cos(x^3)sin(x^2)/(1+x^2) when y=x^2?

 

[Steven G]

Please post your work.

 

[topsquark]

How do you simplify lim cos(x^3)sin(x^2)/(1+x^2) when y=x^2?

What are
[math]\lim_{x \to 0} \dfrac{1}{1 + x^2}[/math]
[math]\lim_{x \to 0} cos(x^3)[/math]
[math]\lim_{x \to 0} sin(x^2)[/math]
-Dan

 

[Elena Baby]

I'm mostly stuck at the simplifying. For example, for the second one, when y=x:



I'm not sure if I simplified this correctly from the part I added the values.
btw I'm not really used to LaTex,so sorry if It looks messy.
Same with approaching the origin with y=x^2.I'm pretty sure this is going to be undefined as most of these examples my professor gave were solved as 'undefined'.

@topsquark also said it was undefined.Is there a simplifying rule I ought to know?

I don't know why the picture I had posted got deleted...



Sin is a bounded function and lim 0*bounded function equals 0,so it'd be equal to 0.I got this one now.

 

[Elena Baby]

What are
[math]\lim_{x \to 0} \dfrac{1}{1 + x^2}[/math]
[math]\lim_{x \to 0} cos(x^3)[/math]
[math]\lim_{x \to 0} sin(x^2)[/math]
-Dan

The first and the second one equal one and the third one equals 0, so the limit would be 0, right?