Lecture Notes: One-Sided Limits

Introduction to One-Sided Limits

• The purpose is to understand one-sided limits both graphically and algebraically.
• Example functions will demonstrate how to evaluate limits.

Example: Evaluating a Piecewise Function

Function Definitions

• Piecewise Function Example:
  • ( f(x) = x + 3 ) if ( x \leq 1 )
  • ( f(x) = x^2 - 2x ) if ( x > 1 )

Goal

• Limit: ( \lim_{{x \to 1^-}} f(x) )
• Determine the limit as ( x ) approaches 1 from the left._

Graphical Understanding

• Graph ( y = x + 3 ):
  • Line with Y-intercept of 3 and slope of 1.
  • Valid for ( x \leq 1 ).
  • Point at ( (1, 4) ) since line is only valid for this region.
• Graph ( y = x^2 - 2x ):
  • Parabola opening upwards.
  • Valid for ( x > 1 ).
  • Open circle at ( (1, -1) ).
• Vertex Calculation:
  • Use (-\frac{b}{2a}) to find vertex at ( x = 1 ).

Evaluating the Limit

• Coming from the left of 1 means ( x ) values are slightly less than 1.
• Plugging values slightly less than 1 into ( x + 3 ) results in Y-values approaching 4.
• Conclusion: ( \lim_{{x \to 1^-}} f(x) = 4 )_

Algebraic Approach

• If function is continuous at the point, plug in the value directly.
• Here, since the function breaks at ( x = 1 ), use the formula valid for ( x \leq 1 ), i.e., ( x + 3 ).
• Result: ( 1 + 3 = 4 )

Right-Sided Limit Example

• Limit: ( \lim_{{x \to 1^+}} f(x) )
• Use ( x^2 - 2x ) for ( x > 1 ).
• As ( x ) approaches 1 from the right, y-coordinates approach -1.
• Conclusion: ( \lim_{{x \to 1^+}} f(x) = -1 )

Full Limit Consideration

• Limit from the left has to equal the limit from the right for overall limit to exist.
• Since ( 4 \neq -1 ), ( \lim_{{x \to 1}} f(x) ) does not exist._

Summary

• Left-sided and right-sided limits can exist independently.
• Overall limit requires equality of left and right limits.
• Graphical understanding aids intuition but algebraic method is more efficient.

Future Implications

• Aim to use formulas without always relying on graphs for efficiency.

This summary synthesizes the key points on evaluating one-sided limits using both graphical and algebraic approaches with a focus on understanding the behavior at specific points.