Limits and Continuity

Definition of a Limit

The limit of a function as x approaches a value describes the behavior of the function near that value, not necessarily at the value itself.

• Notation: means as x approaches a, f(x) approaches L.

• One-sided limits: (from the left), (from the right).

• Existence: The limit exists if and only if both one-sided limits exist and are equal.

Evaluating Limits

• Direct substitution: If is defined and continuous, .

• Factoring and simplifying: Used when direct substitution yields an indeterminate form (e.g., ).

• Special limits: and .

Continuity

A function is continuous at if:

• is defined

• exists

Types of Discontinuities

• Removable discontinuity: The limit exists, but is not defined or not equal to the limit.

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

• Infinite discontinuity: The function approaches infinity near .

Piecewise Functions and Continuity

For piecewise functions, check continuity at the points where the formula changes by evaluating left and right limits and the function value.

• Example:

• Check , , and .

Definition of the Derivative

Derivative as a Limit

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. It is defined as:

• Alternatively,

The derivative represents the instantaneous rate of change of the function at a point.

Finding the Equation of the Tangent Line

• Find using the definition of the derivative.

• Use the point-slope form:

Techniques for Evaluating Limits

Algebraic Manipulation

• Factor numerator and denominator to cancel common terms.

• Rationalize numerator or denominator if radicals are present.

Special Cases and Indeterminate Forms

• Indeterminate forms include and .

• Apply algebraic techniques or L'Hôpital's Rule (if covered).

Examples

• : Factor numerator to get , cancel , and substitute to get $6$.

• : Multiply numerator and denominator by the conjugate to simplify.

Squeeze (Sandwich) Theorem

Statement of the Theorem

If for all near (except possibly at $a$), and , then .

• Used to evaluate limits of functions that are difficult to compute directly.

Graphical Analysis of Limits and Continuity

Using Graphs to Find Limits

• Observe the behavior of the function as approaches a value from both sides.

• Identify discontinuities and classify them as removable, jump, or infinite.

Practice Problems and Solutions

Evaluating Limits Analytically

• Practice with rational, radical, and piecewise functions.

• Check for continuity and differentiability at given points.

Determining Continuity for All Values

• Find values of parameters that make piecewise functions continuous everywhere.

Summary Table: Types of Discontinuities

Additional info: These notes cover foundational concepts in Calculus I, including limits, continuity, the definition of the derivative, and the Squeeze Theorem, with examples and graphical interpretations to reinforce understanding.