A particularly nasty delta-epsilon

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In summary, to prove the function f(x,y) = x/y is continuous, one must show that the limit of f(x,y) as (x,y) approaches (x0,y0) exists and is equal to f(x0,y0). To do this, one can use the fact that for any (x0,y0) with y0 not equal to 0, the absolute value of (x/y - x0/y0) can be written as (|y0x-x0y|/|yy0|), and by manipulating the numerator, the fraction can be made small while keeping the denominator away from 0.
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ssayan3
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Homework Statement



Prove the function f(x,y) = x/y is continuous.
As an added stipulation, the quotients of limits theorem may not be used.

Homework Equations


The Attempt at a Solution


I have absolutely no idea how to go about this one. I can't even get a start on the scratchwork... Can anyone give me any hints to push me along the right way?
 
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  • #2
Here's a start. Pick (x0,y0) with y0 not equal zero.

[tex]\left |\frac x y - \frac {x_0} {y_0}\right |= \left |\frac{y_0x-x_0y}{yy_0}\right |[/tex]

Now subtract and add y0x0 in the numerator and see if you can make the fraction small. You have to get the numerator small and keep the denominator away from 0.
 

Related to A particularly nasty delta-epsilon

1. What is "A particularly nasty delta-epsilon"?

"A particularly nasty delta-epsilon" is a mathematical term used in the field of calculus, specifically in the context of limits. It refers to a specific type of limit that can be challenging to solve due to its complicated nature.

2. How is "A particularly nasty delta-epsilon" different from a regular limit?

The main difference between "A particularly nasty delta-epsilon" and a regular limit is the complexity of the limit. While regular limits can often be solved using basic algebraic techniques, "A particularly nasty delta-epsilon" limits often require more advanced mathematical concepts and techniques.

3. Why is "A particularly nasty delta-epsilon" important in calculus?

"A particularly nasty delta-epsilon" is important in calculus because it helps us understand the behavior of functions near a specific point, known as the limit point. By studying these types of limits, we can determine the continuity and differentiability of functions, which are crucial concepts in calculus.

4. How do you solve "A particularly nasty delta-epsilon" limits?

Solving "A particularly nasty delta-epsilon" limits often involves using a combination of algebraic manipulation, logical reasoning, and advanced mathematical concepts such as the epsilon-delta definition of limits. It requires a thorough understanding of calculus and strong problem-solving skills.

5. Are there any tips for solving "A particularly nasty delta-epsilon" limits?

One helpful tip for solving "A particularly nasty delta-epsilon" limits is to break down the problem into smaller, more manageable parts. Another strategy is to try different approaches and see which one leads to a solution. It is also crucial to have a solid understanding of the underlying concepts and techniques used in solving these types of limits.

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