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 that value. Formally,

• If , then as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L.

Key Properties:

• Limits may exist even if the function is not defined at that point.

• Limits can be evaluated using direct substitution, factoring, rationalization, or L'Hospital's Rule (if indeterminate forms arise).

Example: can be evaluated by factoring numerator and canceling common terms.

Continuity

A function is continuous at a point x = a if:

• f(a) is defined

• exists

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: is discontinuous at x = 1, but the discontinuity is removable.

Average and Instantaneous Rate of Change

Average Speed

The average speed of an object over an interval [a, b] is given by:

where h(t) is the position function.

Instantaneous Speed (Derivative)

The instantaneous speed at time t = a is the derivative of the position function at that point:

This is the slope of the tangent line to the curve at t = a.

Tangent Line Equation

The equation of the tangent line to y = f(x) at x = a is:

Example: For , the instantaneous speed at t = 1 is , and the tangent line at (1, h(1)) is .

Evaluating Limits: Techniques and Examples

Direct Substitution

• If is continuous at , then .

Factoring and Canceling

• Factor numerator and denominator to cancel common terms before substituting.

Rationalization

• Multiply numerator and denominator by a conjugate to simplify expressions involving square roots.

Special Trigonometric Limits

L'Hospital's Rule

• If yields or , then (if the latter limit exists).

Examples of Limits

Each of these can be solved using the above techniques.

Interval Notation and Continuity

Interval Notation

• Used to describe sets of x-values where a function is continuous.

• Example: (a, b) means all x between a and b, not including endpoints.

Describing Continuity

• Identify intervals where the function is defined and has no discontinuities.

• Use limits to check for removable or non-removable discontinuities.

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if f(x) is continuous on [a, b] and k is any value between f(a) and f(b), then there exists c in (a, b) such that f(c) = k.

• Used to show the existence of roots in an interval.

Example: For , check if has a solution in [2, 3] by evaluating f(2) and f(3).

Absolute Value Inequalities and Limits

Absolute Value Inequality

• On [-1, 1],

• Use this to evaluate limits involving absolute values and trigonometric functions.

Example: Use the squeeze theorem to evaluate .

Summary Table: Types of Discontinuities

Key Formulas

• Average rate of change:

• Instantaneous rate of change (derivative):

• Tangent line:

Additional info: The above notes are based on the exam questions, which cover fundamental Calculus I topics such as limits, continuity, average and instantaneous rates of change, tangent lines, and the Intermediate Value Theorem. The examples and explanations are expanded for clarity and completeness.