Limits and Their Evaluation

Definition and Properties of Limits

The concept of a limit is fundamental in calculus, describing the behavior of a function as its input approaches a particular value. Limits are used to define derivatives, continuity, and to analyze functions near points of interest.

• Limit of a Function: The value that a function approaches as the input approaches a specific point.

• Notation: denotes the limit of as approaches .

• One-Sided Limits: (from the left), (from the right).

Example:

• Evaluate Substitute directly since the function is polynomial and continuous:

Limits Involving Trigonometric Functions

Limits can also involve trigonometric functions, which may require special techniques such as L'Hospital's Rule or trigonometric identities.

• Example: As approaches 5, the denominator approaches 0, causing the argument of sine to become very large. The limit does not exist in the usual sense because sine oscillates between -1 and 1 as its argument grows without bound.

Limits Involving Indeterminate Forms

Some limits result in indeterminate forms such as or , which require algebraic manipulation or L'Hospital's Rule.

• Example: Simplify the expression: for . Thus, the limit is 2.

Piecewise Functions and Limits

Evaluating Limits for Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. To evaluate limits at points where the definition changes, consider one-sided limits.

• Example: For , find . - Left-hand limit: - Right-hand limit: Since the left and right limits are not equal, the limit does not exist at .

Continuity of Piecewise Functions

A function is continuous at a point if the limit from both sides equals the function's value at that point.

• Interval Notation: Use interval notation to describe where a function is continuous, e.g., means all between and .

• Points of Discontinuity: Occur where the function's definition changes and the left/right limits do not match, or the function is not defined.

Discontinuity and Reasonableness

Types of Discontinuity

Discontinuities can be classified as removable, jump, or infinite discontinuities.

• Removable Discontinuity: The limit exists, but the function is not defined or has a different value at the point.

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

• Infinite Discontinuity: The function approaches infinity near the point.

Reasonableness: A discontinuity is reasonable if it can be explained by the function's definition or behavior, often involving limits.

Limits Involving Square Roots and Squeeze Theorem

Squeeze Theorem

The Squeeze Theorem is used to evaluate limits of functions that are bounded between two other functions whose limits are known.

• Statement: If for all near , and , then .

• Example: Given for , as , both bounds approach 0, so by the Squeeze Theorem:

Slopes of Secants and Tangents

Secant and Tangent Lines

The slope of a secant line through two points on a curve approximates the rate of change between those points. The slope of the tangent line at a point is the instantaneous rate of change, defined by the derivative.

• Secant Slope Formula:

• Tangent Slope (Derivative):

• Equation of Tangent Line:

Example: Find the slope of the tangent to at using the limit definition, then write the equation of the tangent line at .

Summary Table: Types of Discontinuity

Key Formulas and Concepts

• Limit Definition of Derivative:

• Squeeze Theorem: If and , then

• Continuity at a Point: is continuous at if

Additional info: These notes expand upon the exam questions by providing definitions, examples, and context for limits, continuity, and tangent lines, which are core topics in Calculus I.