Limits and Their Evaluation

Definition and Basic Properties

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing continuity, derivatives, and the behavior of 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).

Examples

• Polynomial Function: Solution: Substitute directly since polynomials are continuous everywhere.

• Trigonometric Function: Explanation: As approaches 5, the denominator approaches 0, causing the argument of sine to become very large. The limit does not exist because oscillates between -1 and 1 infinitely often as its argument grows without bound.

• Indeterminate Form: Solution: As approaches 0, the denominator approaches 0, leading to an infinite limit. The function diverges.

Piecewise Functions and Limits

Piecewise functions are defined by different expressions over different intervals. Evaluating limits at points where the definition changes requires checking both one-sided limits.

• Example:

• Limit at : Left-hand limit: Right-hand limit: Conclusion: The limit does not exist at since the left and right limits are not equal.

Limits Involving Rational Functions

• Example: Solution: Substitute : denominator becomes 0, numerator is 1. The function has a vertical asymptote at .

• Example: Solution: Substitute : numerator , denominator . Limit is .

Continuity and Discontinuity

Definition of Continuity

A function is continuous at a point if:

• is defined

• exists

Interval Notation for Continuity

• Describing Continuity: Use interval notation to specify where a function is continuous. For example, may be continuous on but not at or .

Types of Discontinuity

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

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

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

Example

• Piecewise Function: as defined above has a jump discontinuity at .

Reasonableness of Discontinuity

• Reasonable Discontinuity: If the discontinuity can be removed by redefining the function at a point, it is considered reasonable (removable).

• Unreasonable Discontinuity: If the function has a jump or infinite discontinuity, it is not reasonable.

Tangent Lines and Derivatives

Secant and Tangent Lines

The secant line passes through two points on a curve, while the tangent line touches the curve at one point and has the same slope as the curve at that point.

• Slope of Secant Line:

• Slope of Tangent Line:

Finding the Equation of a Tangent Line

• General Formula: The equation of the tangent line at is

• Example: If is given and is computed, substitute into the formula above.

Special Limit Techniques

Squeeze Theorem

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

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

• Example: for . As , , so by the Squeeze Theorem, .

Summary Table: Types of Discontinuity

Key Formulas

• Limit Definition of Derivative:

• Equation of Tangent Line:

• Squeeze Theorem: If and , then