- #31
zenterix
- 774
- 84
The proposition should actually be with a ##\leq##, ie ##a-\delta \leq f(x)##. Is this what you mean?PeroK said:
Very tricky problem from Spivak's Calculus, Ch. 4 & 5: Graphs/Limits
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SUMMARY
The discussion focuses on the limits and behavior of the function defined by replacing all digits in the decimal expansion of a number after the first occurrence of 7 with zeros, as described in Spivak's Calculus, Chapters 4 and 5. The participants analyze the existence of limits for various values of 'a', particularly around points where the decimal expansion ends in 7 followed by repeating 9s, such as 0.7̅9. They conclude that the limit exists for all real numbers except those with a decimal expansion ending in 7̅9, where the left-hand limit equals 0.7 and the right-hand limit equals 0.8, indicating that the limit does not exist at these points.
PREREQUISITES- Understanding of limits in calculus
- Familiarity with decimal expansions and their properties
- Knowledge of epsilon-delta proofs
- Basic concepts of continuity in functions
- Study epsilon-delta proofs in detail to formalize limit arguments
- Explore the properties of decimal expansions, particularly repeating decimals
- Learn about continuity and discontinuity in functions
- Investigate the implications of function definitions on limit behavior
Students of calculus, mathematicians interested in real analysis, and anyone looking to deepen their understanding of limits and function behavior in mathematical contexts.
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