Calculus Lecture: Understanding Limits

Introduction to Limits

• Definition: A limit is the value that a function (f(x)) approaches as the input (x) approaches some value.
• Example Function: ( f(x) = \frac{x^2 - 4}{x - 2} )
• Goal: Find ( \lim_{{x \to 2}} f(x) )_

Evaluating Limits

Direct Substitution

• Attempt 1: Substitute x = 2 directly into the function.
  • Results in ( \frac{0}{0} ), which is undefined.

Approaching the Limit

• Method: Use numbers close to 2, such as 1.9, 1.99.
  • ( f(1.9) = 3.9 )
  • ( f(1.99) = 3.99 )
  • Conclusion: As x approaches 2, f(x) approaches 4.

Algebraic Simplification

• Technique: Factor and simplify the expression.
  • Factor: ( x^2 - 4 = (x+2)(x-2) )
  • Cancel ( x - 2 ) and evaluate ( x + 2 ) at x = 2.
  • Result: 4

Examples and Practice Problems

Problem 1

• Function: ( x^2 + 5x - 4 )
• Limit: ( \lim_{{x \to 3}} (x^2 + 5x - 4) )
• Solution: Use direct substitution.
  • Result: 20_

Problem 2

• Function: ( \frac{x^2 - 8x + 15}{x - 3} )
• Limit: ( \lim_{{x \to 3}} \frac{x^2 - 8x + 15}{x - 3} )
• Solution: Factor the numerator.
  • Factor: ((x-3)(x-5))
  • Cancel ( x-3 ) and evaluate.
  • Result: -2_

Problem 3

• Complex Fraction: ( \frac{1/x - 1/4}{x-4} )
• Limit: ( \lim_{{x \to 4}} )
• Solution: Multiply by common denominator (4x).
  • Simplify and cancel terms.
  • Result: ( -\frac{1}{16} )_

Problem 4

• Function with Square Root: ( \frac{\sqrt{x} - 3}{x - 9} )
• Limit: ( \lim_{{x \to 9}} )
• Solution: Multiply by conjugate.
  • Result: ( \frac{1}{6} )_

Graphical Evaluation of Limits

One-Sided Limits

• Examples:
  • ( \lim_{{x \to -3^-}} f(x) = 1 )
  • ( \lim_{{x \to -3^+}} f(x) = 2 )
  • If left ( \neq ) right, limit does not exist.

Types of Discontinuities

• Jump Discontinuity: Different limits from left and right.
• Removable Discontinuity: Hole in the graph, limit exists but differs from function value.
• Infinite Discontinuity: Vertical asymptote, limit approaches infinity.

Summary

• Evaluating Limits: Techniques include substitution, factoring, and rationalizing.
• Graphical Understanding: Important for visualizing limits and identifying discontinuities.