Limits and Boundary and Func help

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SUMMARY

This discussion focuses on mathematical proofs and limits, specifically addressing how to determine the proximity of the expression 3x + 2 to the value 14 within specified distances. Participants explore the implications of integer properties, proving that if k^3 is even, then k must also be even. Additionally, the irrationality of 3√2 is examined, paralleling the proof of √2's irrationality. The conversation also questions whether the sum of two irrational numbers must be irrational, inviting proofs or counterexamples.

PREREQUISITES
  • Understanding of limits in calculus
  • Knowledge of integer properties and proofs
  • Familiarity with irrational numbers and their properties
  • Basic proof techniques in mathematics
NEXT STEPS
  • Study the concept of limits in calculus, focusing on epsilon-delta definitions
  • Learn about integer properties and proof techniques in number theory
  • Explore the proof of the irrationality of √2 and its implications for other roots
  • Investigate the properties of irrational numbers and their sums, including relevant examples
USEFUL FOR

Students studying mathematics, particularly those tackling proofs in number theory and calculus, as well as educators seeking to enhance their understanding of irrational numbers and limits.

mms6
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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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These look like homework problems. Are they? What have you tried on them?
 

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