Introduction to Limits

Basic Concept of Limits

• Evaluating limits can be done analytically and graphically.

Example 1: Evaluating a Limit

• Function: ( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} )
• Direct Substitution:
  • Plugging in 2 gives ( \frac{0}{0} ) (undefined).
• Evaluating Close Values:
  • Calculate ( f(1.9) ) and ( f(2.1) ) to find the limit approaches 4.
• Factoring Method:
  • Factor ( x^2 - 4 = (x + 2)(x - 2) )
  • Cancel ( x - 2 ) and use direct substitution:
    • ( \lim_{x \to 2} (x + 2) = 4 )

Example 2: Another Limit

• Limit: ( \lim_{x \to 5} (x^2 + 2x - 4) )
• Use direct substitution:
  • ( 5^2 + 2(5) - 4 = 31 )

Example 3: Limit with a Fraction

• Limit: ( \lim_{x \to 3} \frac{x^3 - 27}{x - 3} )
• Direct substitution gives ( \frac{0}{0} )
• Factoring:
  • Use the difference of cubes: ( x^3 - 27 = (x - 3)(x^2 + 3x + 9) )
  • Cancel ( x - 3 ) and substitute:
    • Result: ( 27 )

Example 4: Complex Fraction Limit

• Limit: ( \lim_{x \to 3} \frac{1/x - 1/3}{x - 3} )
• Multiply by common denominator ( 3x ):
  • Simplify and cancel ( x - 3 )
  • Result: ( \lim_{x \to 3} \frac{-1}{9} )

Example 5: Limit with Square Roots

• Limit: ( \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} )
• Multiply by the conjugate to simplify:
  • Result: ( \lim_{x \to 9} \frac{1}{6} )

Example 6: Complex Fraction with Radicals

• Limit: ( \lim_{x \to 4} \frac{1/\sqrt{x} - 1/2}{x - 4} )
• Multiply top and bottom by common denominator and conjugate:
  • Result: ( \lim_{x \to 4} \frac{-1}{16} )

Evaluating Limits Graphically

One-Sided Limits

• Example: ( \lim_{x \to -3} ) from left: ( y = 1 ) (value approaches from left)
• From right: ( y = -3 ) (value approaches from right)
• If left and right limits differ, the limit does not exist.

Continuous vs. Discontinuous Functions

• Closed circle indicates function value.
• Example: ( \lim_{x \to -2} ) both sides equal ( -2 ), so limit exists.
• Jump Discontinuity: If limits differ, it is non-removable.
• Vertical Asymptote: If limit approaches infinity, it is undefined (non-removable).

Conclusion

• Different types of discontinuities:
  • Jump Discontinuity: Non-removable.
  • Hole (Removable): Can be fixed.
  • Infinite Discontinuity: Non-removable (vertical asymptote).