Limits and Direct Substitution

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SUMMARY

The discussion centers on the concept of limits in calculus, specifically addressing the validity of using direct substitution to evaluate limits. The participants highlight that for a limit to exist, both the left-hand limit and right-hand limit must converge to the same value. It is noted that the function f(0) may not be defined or could differ from the limit value of 2, indicating a potential removable discontinuity at x=2. This understanding is crucial for correctly determining the behavior of functions at specific points.

PREREQUISITES
  • Understanding of calculus concepts, specifically limits.
  • Familiarity with left-hand and right-hand limits.
  • Knowledge of removable discontinuities in functions.
  • Basic proficiency in evaluating limits using direct substitution.
NEXT STEPS
  • Study the properties of limits in calculus, focusing on the Squeeze Theorem.
  • Learn about removable and non-removable discontinuities in functions.
  • Practice evaluating limits using both direct substitution and graphical methods.
  • Explore the implications of limits in real-world applications, such as physics and engineering.
USEFUL FOR

Students studying calculus, educators teaching limit concepts, and anyone seeking to deepen their understanding of function behavior at specific points.

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Why wouldn't answer choice "C" be true as well?

[PLAIN]http://i.min.us/idPNiK.png

Homework Equations



I thought that you could use direct substitution on this limit.

The Attempt at a Solution



I did get answer choices A and B. For the limit to exist, both the left and right hand limits have to match up.
 
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f(0) might not be defined or it may be that f(0)≠2. In other words, f(x) might have a removable discontinuity at x=2.
 

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