Restrictions for Limit Existence: Integer Conditions for a,b,c | Homework Help

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In summary, the student is trying to solve a problem and is not sure how to proceed. He is grateful for the help that has been provided.
  • #1
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Homework Statement



For positive integers a,b and c. State a reasonable condition which guarantees the following limit exists:

http://img691.imageshack.us/img691/1923/limit.jpg [Broken]

Homework Equations



Maybe Binomial Theorem (see http://en.wikipedia.org/wiki/Binomial_theorem)
Mayve L'Hopitols (see http://en.wikipedia.org/wiki/L'Hôpital's_rule)

The Attempt at a Solution



I am really stuck with this, I have spent the whole day doing excercises and putting this one off,

In lectures we were told to consider the behaviour of the limit along lines through the x axis:

So Consider y=αx (α= alpha, not 'a' as in 'a' from the limit) for α an eliment of Real Numbers

I substituted this in and got:

http://img27.imageshack.us/img27/9015/limmo.jpg [Broken]
THAT Y IS SUPPOSED TO BE APLHA! SORRY!

Ive spoken to people who have said I need to apply the binomial expansion at some point in the question. I am not sure I am even supposed to substitute in y=αx.

I am supposed to end up with something like a+c > bAny help is massively appreciated
 
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  • #2
In this type of problem, the usual way the limit fails to exist is that the value depends on the direction from which you approach zero. If we consider approaching along the x- or y-axis, the expression simplifies greatly, and we can get some initial results.

Suppose we're approaching zero along the x-axis. Then the limit simplifies to [tex]x^{a-2c}[/tex], and if we're approaching along the y-axis, we have [tex]y^{b-2c}[/tex].

The first expression (as x goes to 0) goes to infinity if a<2c, to 1 if a=2c, and to zero if a>2c. The second has the same behavior, depending on the relationship of b to 2c.

So if a and b don't have the same relationship to 2c, the limit does not exist. This is a necessary condition, but may not be sufficient. This should give you an idea of how to attack it.
 
  • #3
Ok, thank you that helps a lot

could I say:

[(x^2)+(y^2)]^c is greater than [(x^2)*(y^2)]^c

The numerator needs to be smaller thatn the denomonator so 2c > a + b

if the numerator is smaller then the limit tends to 0 but if it is bigger it will tend to infinity.

But I am pretty sure this will not get me the marks, i don't know.

I think i need to try your approuch of thinking about the limit along the x and y-axis etc. Its 3am now i think il sleep and do it tomoro! thanks for the help
 

1. What are restrictions for limit existence and how do integer conditions for a, b, and c play a role?

Restrictions for limit existence refer to the conditions that must be met in order for a mathematical limit to exist. These restrictions become more complex when dealing with integers a, b, and c, as they must also follow certain conditions in order for the limit to exist.

2. What is the importance of integer conditions in determining limit existence?

Integer conditions are important because they add an additional layer of complexity to determining limit existence. Without satisfying these conditions, the limit may not exist or may have a different value.

3. What are some common integer conditions for a, b, and c in limit existence?

Some common integer conditions for a, b, and c include avoiding division by zero, maintaining the same sign for both the numerator and denominator, and avoiding any non-integer values in the limit expression.

4. How can I check if the integer conditions for a, b, and c are satisfied in a limit problem?

To check if the integer conditions are satisfied, you can plug in the values of a, b, and c into the limit expression and simplify. If the expression is undefined or contains non-integer values, then the conditions are not satisfied.

5. Can the integer conditions for a, b, and c ever be ignored in limit problems?

No, the integer conditions must always be taken into consideration in limit problems. Ignoring them can lead to incorrect or undefined results.

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