Introduction to Limits

Overview

• Understanding limits and evaluating them both analytically and graphically.

Example Problems and Solutions

Problem 1: Limit as x approaches 2

• Function: ( f(x) = \frac{x^2 - 4}{x - 2} )
• Direct Substitution:
  • Substitute 2: ( 2^2 - 4 = 0 ) -> ( \frac{0}{0} ) is undefined.
• Approaching 2:
  • Calculate ( f(2.1) ): ( \frac{0.41}{0.1} ) -> 4.1
  • Calculate ( f(2.01) ): ( \frac{0.0401}{0.01} ) -> 4.01
  • Conclusion: As x approaches 2, the limit is 4.
• Alternative Approach:
  • Factor: ( x^2 - 4 = (x+2)(x-2) )
  • Cancel ( x - 2 ), evaluate ( x + 2 ) at x = 2 -> Limit is 4.

Problem 2: Limit as x approaches 5

• Function: ( x^2 + 2x - 4 )
• Direct Substitution:
  • Substitute 5: ( 5^2 + 2 \times 5 - 4 = 31 )
• Conclusion: Limit is 31.

Problem 3: Limit as x approaches 3

• Expression: ( \frac{x^3 - 27}{x - 3} )
• Direct Substitution:
  • Results in ( \frac{0}{0} ).
• Factor the Expression:
  • Use difference of cubes formula: ( x^3 - 27 = (x-3)(x^2 + 3x + 9) )
  • Cancel ( x - 3 ), substitute 3 in remaining expression.
  • Conclusion: Limit is 27.

Problem 4: Complex Fraction

• Expression: ( \frac{1/x - 1/3}{x-3} )
• Simplification:
  • Multiply numerator and denominator by 3x to eliminate fractions.
  • Factor and simplify to find the limit.
  • Conclusion: Limit is ( -\frac{1}{9} ).

Problem 5: Square Roots

• Expression: ( \frac{\sqrt{x} - 3}{x-9} )
• Rationalization:
  • Multiply by conjugate ( \sqrt{x} + 3 ).
  • Simplify and cancel terms, then substitute.
  • Conclusion: Limit is ( \frac{1}{6} ).

Problem 6: Complex Fraction with Radicals

• Process:
  • Multiply by common denominator and conjugate.
  • Factor and cancel terms, then substitute.
  • Conclusion: Limit is ( -\frac{1}{16} ).

Graphical Evaluation of Limits

Evaluating Graphically

• Consider the graph of f(x) and discuss limits approaching from left and right.
• Example: Limit as x approaches -3 from left/right:
  • From left: y-value is 1.
  • From right: y-value is -3.
  • Limit does not exist if left and right limits are not equal.

Discontinuities and Asymptotes

• Types of Discontinuities:
  • Jump Discontinuity: Non-removable, e.g., at x = -3.
  • Hole (Removable Discontinuity): e.g., at x = -2.
  • Infinite Discontinuity: Near vertical asymptotes.
• Example of Infinite Discontinuity:
  • Function: ( \frac{1}{x-3} ) has vertical asymptote at x=3.

Conclusion

• Understanding limits, various methods to evaluate them, and identifying types of discontinuities in graphs.