Limits, Continuity, Rates of Change, and Derivatives

Average Rate of Change

The average rate of change of a function measures the average change in output per unit of input over an interval. It is geometrically represented as the slope of the secant line connecting two points on the graph.

• Formula:

• Points: and

• Interpretation: Use two x-values; do not confuse with instantaneous rate.

• Secant Slope: Same formula as average rate of change.

Example: For on , , , so average rate is .

Limits and Limit Laws

A limit describes the value a function approaches as the input approaches a specific point. It is foundational for calculus concepts such as continuity and derivatives.

• Notation:

• Direct Substitution: Try first; if it yields a real number, that's usually the limit.

• Indeterminate Form (0/0): Requires algebraic manipulation (factoring, rationalizing, combining fractions).

• Limit Laws:

  • Sum:

  • Difference:

  • Constant Multiple:

  • Product:

  • Quotient: (if denominator limit )

• Important Trigonometric Limits:

Example: by direct substitution.

Factoring Example: : Factor numerator, cancel , substitute , answer .

Rationalizing Example: : Multiply by conjugate, simplify, answer .

One-Sided Limits

One-sided limits consider approaching a point from only one direction (left or right). The two-sided limit exists only if both one-sided limits are equal.

• Left-hand limit: (approach from values less than )

• Right-hand limit: (approach from values greater than )

• Two-sided limit: Exists only if left and right limits are equal.

• Graphical Interpretation: Trace from left and right; filled dot may represent , not the limit; a hole can have a limit.

Example: If and , then .

Limits Involving Infinity

Limits can involve infinity in two main ways: as the function approaches a finite -value (infinite limits), or as approaches infinity (end behavior).

• Infinite Limits: or ; function grows without bound near (vertical asymptote).

• Limits as or : Describes end behavior, especially for rational functions.

• Degree Comparison for Rational Functions:

  • Top degree < bottom degree: limit

  • Same degree: limit ratio of leading coefficients

  • Top degree > bottom degree: generally no finite limit; inspect leading terms

Examples:

Continuity

A function is continuous at if there is no break, hole, or jump at that point. Three conditions must be satisfied:

• exists

• exists

Types of Discontinuity:

• Removable: Hole; limit exists but is missing or incorrect.

• Jump: Left and right limits exist but are different.

• Infinite: Function becomes unbounded (vertical asymptote).

Common Continuity Facts:

• Polynomials: continuous everywhere

• Rational functions: continuous where denominator

• Root functions: continuous where defined

• Sine and cosine: continuous everywhere

Piecewise Procedure: Find , left-hand limit, right-hand limit, compare, then check if limit equals .

Instantaneous Rate of Change

The instantaneous rate of change at a point is the rate at one exact input value. It is the slope of the tangent line at that point and is given by the derivative.

• Formula:

• Obtained by taking the limit of average rates as the interval shrinks to a single point.

Formal Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero.

• General:

• At a point:

• Steps:

  1. Find

  2. Subtract

  3. Expand and simplify

  4. Factor if possible

  5. Cancel

  6. Take the limit as

Note: Do not substitute before simplifying; this creates .

Derivative as Tangent Slope

If a function is differentiable at , then is the slope of the tangent line at . The tangent line equation uses this slope.

• Tangent Line Formula: , where

Example: For at , . At , . Tangent line: or .

Basic Differentiation Rules

Several fundamental rules allow quick computation of derivatives for common functions.

• Constant Rule:

• Power Rule:

• Constant Multiple Rule:

• Sum/Difference Rule: Differentiate term-by-term

• Trigonometric Derivatives:

Example: , . At , .

Problem Recognition Table

The following table summarizes common question types, their meaning, and recommended strategies:

Common Exam Traps

• is not the answer; it signals indeterminate form.

• and can be different.

• Two-sided limit requires both sides to agree.

• Continuity requires limit equals .

• Average rate uses two points; instantaneous rate uses one.

• .

• Power rule: multiply by exponent, subtract 1 from exponent.

• ; negative exponents mean reciprocals.

• For graph questions, follow the curve, not just the filled dot.

Formula Sheet

• Average rate:

• Derivative:

• Derivative at a:

• Tangent line: ,

• Continuity: exists; limit exists; limit

• Two-sided limit: Exists only if left right

• Power rule:

Practice Problems (with Answers)

• Evaluate : 9

• Evaluate : 6

• Evaluate : 1/4

• Left limit at is 4 and right limit is 4. Find the two-sided limit: 4

• Left limit at is 4 and right limit is 6. Find the two-sided limit: DNE

• but . Is continuous at 2? No

• Find the average rate of change of on : 5

• Find for :

• Find the tangent slope of at : -27

• Write the formal definition of :

• For , find the slope at and the tangent line: Slope ;

• Evaluate :

Additional info: This guide covers the foundational concepts for Calculus I Unit 1, including limits, continuity, rates of change, and derivatives, as outlined in the course syllabus. It is suitable for exam preparation and self-study.