Limit Definition of Derivative

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SUMMARY

The limit definition of the derivative was applied to the piecewise function f(x) = { sqrt(x^2+1) if x<=0, 0 if x>0. The function is not continuous at x=0, leading to its non-differentiability at that point. The left-hand limit approaches 1 while the right-hand limit approaches 0, confirming the discontinuity. This discrepancy in limits illustrates the importance of continuity in differentiability.

PREREQUISITES
  • Understanding of limit definitions in calculus
  • Knowledge of piecewise functions
  • Familiarity with continuity and differentiability concepts
  • Basic graphing skills for visualizing functions
NEXT STEPS
  • Study the formal definition of continuity in calculus
  • Learn about the implications of discontinuities on differentiability
  • Explore limit calculations for piecewise functions
  • Practice finding derivatives using the limit definition with various functions
USEFUL FOR

Students studying calculus, particularly those focusing on derivatives and continuity, as well as educators seeking to clarify these concepts in a teaching context.

cjaylee
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Homework Statement


Use the limit definition of derivative to determine the derivative of the following function:

f(x) = { sqrt(x^2+1) if x<=0
0 if x>0

Homework Equations



I'm not sure as to why the function is not continuous at x=0, and so it's not differentiable at that point.

The Attempt at a Solution



The left-hand limit and right-hand limit give me 0. And if I plot a graph, the graph hits 0 as it moves from one function to the other.
 
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The left hand limit is "1", not "0"! I would think that would be obvious. What is f(-0.001)?
 

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