Limits and Their Evaluation

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for studying continuity, derivatives, and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets arbitrarily close to a.

• Notation:

• Key Properties:

  • If the left-hand and right-hand limits are equal, the limit exists.

  • Limits can be finite, infinite, or may not exist.

Example: Evaluating Limits

• Algebraic Manipulation: Factorization, rationalization, or substitution may be used to evaluate limits.

• Example:

  • Divide numerator and denominator by :

• Example:

  • Divide numerator and denominator by :

• Example:

  • As , dominates denominator, numerator is bounded:

  • Limit is $0$.

Asymptotes of Functions

Horizontal and Vertical Asymptotes

Asymptotes are lines that a graph approaches but never touches. They help describe the end behavior and undefined points of functions.

• Horizontal Asymptotes: Found by evaluating and .

• Vertical Asymptotes: Occur where the function becomes unbounded, typically where the denominator is zero and the numerator is nonzero.

Example: Finding Asymptotes

• Function:

• Horizontal Asymptote:

  • As ,

  • As , (no horizontal asymptote in this form)

  • Additional info: For rational functions, horizontal asymptotes are found by comparing degrees of numerator and denominator.

• Vertical Asymptote:

  • Set denominator ; has no real solution, so no vertical asymptote.

Continuity and Discontinuity

Definition of Continuity

A function is continuous at a point if its limit exists at that point and equals the function's value there. Discontinuities occur when this condition fails.

• Types of Discontinuity:

  • Jump Discontinuity: Left and right limits exist but are not equal.

  • Infinite Discontinuity: Function approaches infinity at the point.

  • Removable Discontinuity: Limit exists, but function is not defined or not equal to the limit.

Example: Piecewise Function Discontinuity

• Function:

  x	f(x)
  x < 0
  0 \leq x < 3
  x \geq 3

• Discontinuity Points:

  • At : is not defined for (real numbers), so discontinuity at .

  • At : Check if and match at :

    Jump discontinuity at .

Continuity of Composite Functions

Continuous Functions and Limits

If and are continuous at a point, their sum, difference, product, and quotient (if denominator is nonzero) are also continuous at that point.

• Example: If and , find .

  • Since and are continuous, and .

Making a Function Continuous

Removing Discontinuity

To make a function continuous at a point, redefine its value at that point to match the limit as x approaches the point.

• Example: at

  • Factor numerator:

  • So for

  • To make continuous at , define

Summary Table: Types of Discontinuity