Introduction to Limits

• Explore basic concepts of limits.
• Focus on evaluating limits analytically and graphically.

Example Problem: Limit as x Approaches 2

• Function: ( \frac{x^2 - 4}{x - 2} )
• Direct Substitution:
  • Plugs in 2; results in ( \frac{0}{0} ) (undefined).
• Plugging Nearby Values:
  • Using ( x = 2.1 ), ( x = 2.01 ) shows limit approaches 4.
• Factoring Method:
  • Factor ( x^2 - 4 ) as ( (x+2)(x-2) )
  • Cancel ( x-2 ) and apply direct substitution.
  • Limit is 4.

Evaluating Limits: Analytical Techniques

Direct Substitution

• Example: Limit of ( x^2 + 2x - 4 ) as ( x \to 5 ).
  • Substitute and simplify: Result is 31.

Factoring

• Example: Limit of ( \frac{x^3 - 27}{x - 3} ) as ( x \to 3 ).
  • Factor using difference of cubes.
  • Result after cancellation and substitution is 27.

Complex Fractions

• Example: Limit of ( \frac{1/x - 1/3}{x - 3} ) as ( x \to 3 ).
  • Multiply by common denominator, simplify, and result is ( -\frac{1}{9} ).

Radicals

• Example: Limit of ( \frac{\sqrt{x} - 3}{x - 9} ) as ( x \to 9 ).
  • Multiply by conjugate, cancel, and result is ( \frac{1}{6} ).

Graphical Evaluation of Limits

• Concept: Evaluate limits by identifying y-values on graphs as x approaches a number.
• One-Sided Limits:
  • Approach from left or right of the number.
  • Example: As ( x \to -3 ), left limit is 1, right limit is -3.

Discontinuities

• Jump Discontinuity:
  • A break in the graph (e.g., at ( x = -3 )).
• Hole (Removable Discontinuity):
  • Point missing in the graph (e.g., at ( x = -2 )).
• Infinite Discontinuity:
  • Graph goes to infinity (e.g., vertical asymptote at ( x = 3 )).

Additional Graphical Examples

• Limits at Various Points:
  • Approach from left/right and evaluate function values at specific points.
• Vertical Asymptotes:
  • Example with function ( \frac{1}{x-3} ). Results in infinity as x approaches 3.

Key Takeaways

• Use multiple techniques to evaluate limits: direct substitution, factoring, simplifying complex fractions, and multiplying by conjugates.
• Graphs can provide visual insights into limits, one-sided limits, and discontinuities.