Limits and Boundary and Func help

In summary, the conversation and problems revolve around determining the proximity of a value x to 4 in order to satisfy certain conditions for the equation 3x + 2 = 14. Part (a) requires proving that if a cube of any integer is even, then the integer itself must be even. Part (b) involves proving that 3√2 is irrational using the same method as the proof for √2 and utilizing the result from part (a). Lastly, part (c) asks whether the sum of two irrational numbers must also be irrational and requires an appropriate proof or counterexample.
  • #1
mms6
1
0
How close to 4 do we have to take x so that 3x + 2 is within a distance of (a) 0.1 and (b) 0.01 from 14?

http://i.imagehost.org/0222/Picture_1.png



a) Let k be any integer. Prove that if k^3 is even, then k is even.

b) Prove that 3√2 is irrational. Hint: Mimic the proof that √2 is irrational and apply the result from part (a).

c) Suppose two numbers a and b are irrational. Must the sum a + b be irrational? Justify your answer with an appropriate proof or counterexample


Thanks guys!
 
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  • #2
These look like homework problems. Are they? What have you tried on them?
 

1. What is the difference between a limit and a boundary?

A limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. A boundary, on the other hand, refers to the values at which a function changes behavior or value. In other words, a limit is a value that a function approaches, while a boundary is a value that a function cannot reach.

2. How do limits and boundaries affect the behavior of a function?

Limits and boundaries play a crucial role in determining the behavior of a function. They help us understand how a function behaves near a certain point or as the input approaches a particular value. They also help us identify any discontinuities or changes in behavior of a function.

3. Can limits and boundaries be used to prove the existence of a function?

Yes, limits and boundaries can be used to prove the existence of a function. If a function has a limit at a certain point and the limit exists, then it is guaranteed that the function exists at that point as well.

4. How can understanding limits and boundaries help in solving real-world problems?

Limits and boundaries are used in many real-world applications, such as predicting the behavior of a stock market, modeling population growth, or calculating rates of change. Understanding these concepts can help in making accurate predictions and solving various problems in different fields.

5. Are there any techniques to evaluate limits and boundaries?

Yes, there are various techniques to evaluate limits and boundaries, such as using algebraic manipulation, substitution, and L'Hopital's rule. It is essential to understand these techniques in order to accurately evaluate limits and boundaries in more complex functions.

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