Limits and Continuity

Evaluating Limits

Limits are fundamental to calculus, describing the behavior of functions as inputs approach a particular value. They are essential for defining derivatives and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is written as .

• Key Properties:

  • If , then .

  • Limits may not exist if the left and right limits differ or if the function grows without bound.

• Example:

Special Relativity and Limits

In physics, limits are used to analyze formulas such as the Lorentz contraction in special relativity:

• Lorentz Contraction Formula:

  • Where is the proper length, is the velocity of the object, and is the speed of light.

• Application: As approaches , the length contracts toward zero.

Indeterminate Forms and L'Hospital's Rule

Some limits result in indeterminate forms such as or . L'Hospital's Rule can be used to evaluate these limits:

• L'Hospital's Rule: If yields or , then:

$

• Example:

Continuity and Discontinuity

Definition of Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value there.

• Mathematical Definition: is continuous at if .

• Types of Discontinuities:

  • Removable: The limit exists, but the function is not defined or is defined differently at that point.

  • Jump: The left and right limits exist but are not equal.

  • Infinite: The function approaches infinity near the point.

Example Function Analysis

• Given :

  • Domain: Values of for which the expression under the square roots is non-negative.

  • Discontinuities: Points where the function is not defined or not continuous.

Intermediate Value Theorem (IVT)

Statement of the Theorem

The IVT guarantees that for any continuous function on a closed interval, every value between the function's values at the endpoints is achieved at some point within the interval.

• Formal Statement: If is continuous on and is between and , then there exists such that .

• Application: Used to prove the existence of roots or solutions within an interval.

Extra Credit: Functions Defined by Rationality

Piecewise Functions and Density of Rationals/Irrationals

Some functions are defined differently for rational and irrational numbers, illustrating the density of these sets in the real numbers.

• Example Function:

  • if is irrational

  • if is a rational number in lowest terms

• Graphical Behavior: The function is discontinuous at every rational point except possibly at .

• Hint: Between any two real numbers, there are both rational and irrational numbers, so the graph is dense with points at different heights for rationals and zero for irrationals.

Summary Table: Types of Discontinuities