What does this Limit mean geometrically?

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In summary, the conversation discusses finding limits at a point and determining if the limit exists by considering different paths. It also mentions that if the limit is independent of a variable, it means that the limit exists. Additionally, it is mentioned that checking along different paths is not the only way to see if a limit exists.
  • #1
evinda
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Hello! (Wave)

I want to find the following limit, if it exists.

$\lim_{(x,y) \to (0,0)} \frac{\cos x-1-\frac{x^2}{2}}{x^4+y^4}$ If we say : let $(x,y) \to (0,0)$ along the line $y=0$ , what exactly does it mean geometrically?

Also, if we want to check whether the limit $\lim_{(x,y) \to (0,0)} g(x,y) $ exists where $g(x,y)=\frac{xy}{x^2+y^2}$, we pick $y=mx$ and since $g(x,mx)$ depends on $m$ we deduce that the limit does not exist.

If $g(x,mx)$ would be independent on $m$, would that mean that the limit exists? If so, why?
 
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  • #2
evinda said:
Hello! (Wave)

I want to find the following limit, if it exists.

$\lim_{(x,y) \to (0,0)} \frac{\cos x-1-\frac{x^2}{2}}{x^4+y^4}$ If we say : let $(x,y) \to (0,0)$ along the line $y=0$ , what exactly does it mean geometrically?

Also, if we want to check whether the limit $\lim_{(x,y) \to (0,0)} g(x,y) $ exists where $g(x,y)=\frac{xy}{x^2+y^2}$, we pick $y=mx$ and since $g(x,mx)$ depends on $m$ we deduce that the limit does not exist.

If $g(x,mx)$ would be independent on $m$, would that mean that the limit exists? If so, why?

What do you mean by "independent on m"?
 
  • #3
Prove It said:
What do you mean by "independent on m"?

I mean that the result of the limit isn't a function of $m$...
 
  • #4
Also, if we want to check if the limit $\lim_{(x,y) \to (0,0)} \frac{\sin{(2x)}-2x+y}{x^3+y}$ exists, we can consider the limit along the line $y=0$ and the limit along $x=0$ and we will see that they are not equal. Is this the only way to see that the limit does not exist?
 

FAQ: What does this Limit mean geometrically?

What does this Limit mean geometrically?

The limit of a function at a certain point can be thought of as the y-value that the function approaches as the x-value approaches the given point on the graph. It represents the behavior of the function near that point.

How do you visualize a Limit?

A limit can be visualized by looking at the graph of the function. The point on the x-axis where the limit is being evaluated is marked and the behavior of the function as it approaches that point is observed. This can help in understanding the limit conceptually.

What is the difference between a Limit and a value of a function?

A limit is a theoretical concept that represents the behavior of a function near a certain point. It may or may not be equal to the value of the function at that point. The value of a function, on the other hand, is the actual y-value of the function at a given x-value.

Can a Limit be undefined?

Yes, a limit can be undefined if the behavior of the function near the given point is not defined. This can happen when there is a vertical asymptote, a hole, or a jump in the graph of the function at that point.

How do you find the Limit of a function?

The limit of a function can be found by evaluating the function as the x-value approaches the given point. If the resulting values approach a certain number, that number is the limit. If the values do not approach a certain number, the limit may be undefined.

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