Introduction to Limits

Basic Concepts

• Limits: Understanding limits is fundamental to calculus. They help us examine the behavior of functions as they approach a specific point.
• Evaluating Limits: Limits can be evaluated both analytically and graphically.

Analytical Evaluation of Limits

Direct Substitution

• Example: Limit as x approaches 2 of ( \frac{x^2 - 4}{x - 2} )
  • Direct substitution leads to ( \frac{0}{0} ), which is undefined.
  • Use values close to 2 to approximate the limit.
  • As x approaches 2, the limit approaches 4.

Factoring Method

• Example: Factor ( x^2 - 4 = (x+2)(x-2) ).
  • Cancel ( x-2 ) after factoring to avoid zero in the denominator.
  • Limit becomes ( x+2 ) as x approaches 2, resulting in a limit of 4.

Polynomial Functions

• Example: Limit as x approaches 5 of ( x^2 + 2x - 4 ).
  • Direct substitution is possible; the result is 31.

Rational Functions

• Example: Limit as x approaches 3 of ( \frac{x^3 - 27}{x - 3} ).
  • Factor as a difference of cubes and cancel terms.
  • Substitute to find the limit as 27.

Complex Fractions

• Example: Limit as x approaches 3 of ( \frac{1/x - 1/3}{x - 3} ).
  • Use common denominators and multiplying by conjugates.
  • Resulting limit is (-\frac{1}{9}).

Square Roots and Conjugates

• Example: Limit as x approaches 9 of ( \frac{\sqrt{x} - 3}{x - 9} ).
  • Multiply by the conjugate and simplify.
  • Resulting limit is ( \frac{1}{6} ).

Evaluating Limits Graphically

One-Sided Limits

• Approach: Determine the y-value as x approaches from left or right.
• Example: Limit as x approaches -3.
  • From the left, limit is 1.
  • From the right, limit is -3.
  • If limits from both sides differ, the limit does not exist.

Function Values

• Closed Circle: Indicates the function value at a point (e.g., ( f(-3) = -3 )).

Additional Graphical Scenarios

• Jump Discontinuity: Points where the graph jumps; limits from either side are different.
• Removable Discontinuity (Hole): Can be "removed" by rewriting the function.
• Infinite Discontinuity: Typically occurs at vertical asymptotes (e.g., ( \frac{1}{x-3} )).

Conclusion

• Understanding both analytical and graphical approaches is crucial for evaluating limits.
• Different types of discontinuities help identify behavior at specific points.