Limit Existence and the Epsilon-Delta Proof

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In summary, a limit is a fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value. It exists when the function approaches the same value regardless of direction, but does not exist when the function does not approach a well-defined value. A limit cannot exist at a discontinuity, but can exist on either side of it. In real-world applications, limits are used to model and predict real-world phenomena, such as determining maximum or minimum values in optimization problems.
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faradayscat
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Homework Statement


Prove that

lim (x,y,z)→(0,0,0) 2xz/(x²+y²+z²) = 0

Homework Equations


My teacher wants me to show this using epsilon delta, so

0<√(x²+y²+z²)<∂ ⇒ |f(x,y,z) - 0| < ε

The Attempt at a Solution


The limit does not exist apparently.. when you approach the limit along different paths you get different answers.. say, along (t,0,t) gives 1 while along (0,0,t) gives 0. Did my professor make a mistake in this assignment, or does the limit actually exist and I'm missing something?
 
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You're right. Your professor made a mistake.
 
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FAQ: Limit Existence and the Epsilon-Delta Proof

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value. It is used to determine the value that a function is approaching, rather than the value it is actually at.

2. How do you know if a limit exists or not?

A limit exists if the function approaches the same value regardless of which direction the input approaches the limit. In other words, the limit exists if the function has a well-defined value as the input gets closer and closer to the limit.

3. What does it mean if a limit does not exist?

If a limit does not exist, it means that the function does not approach a single, well-defined value as the input approaches the limit. This can happen for various reasons, such as the function having a vertical asymptote or oscillating infinitely between two values.

4. Can a limit exist at a discontinuity?

No, a limit cannot exist at a discontinuity. This is because a discontinuity means that the function is not continuous at that point, and a limit only exists for continuous functions. However, it is possible for a limit to exist on either side of a discontinuity.

5. How is the limit used in real-world applications?

The concept of a limit is used in various fields such as physics, engineering, and economics to model and predict real-world phenomena. For example, it is used in calculus to determine the maximum or minimum values of a function, which can be applied in optimization problems in engineering and economics.

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