Section 2.1 – The Idea of Limits

Review of Slope and Lines

Calculus begins with concepts from algebra, such as finding the slope and equation of a line. These ideas are foundational for understanding limits, average velocity, and the slope of tangent lines.

• Slope Formula: The slope m between two points and is given by:

• Equation of a Line: The point-slope form is:

• Example: For points (2, 4) and (4, 16): Equation:

Introduction to Limits

A limit describes the value that the outputs of a function approach as the inputs approach a given value. This concept is central to calculus, especially in defining instantaneous velocity and the slope of a tangent line.

• Limit Notation:

• Secant Line: A line passing through two points on a graph; its slope represents average velocity.

• Tangent Line: A line touching the graph at one point; its slope represents instantaneous velocity.

Average and Instantaneous Velocity

Average velocity is calculated over an interval, while instantaneous velocity is the limit as the interval shrinks to a single point.

• Average Velocity Formula:

• Instantaneous Velocity: The limit of average velocity as :

• Example: Using a table of positions, as the interval shrinks, the average velocity approaches the instantaneous velocity.

Section 2.2 – Definition of Limits

Informal and One-Sided Limits

Limits can be approached from the left or right. The notation distinguishes these cases:

• Right-Sided Limit:

• Left-Sided Limit:

• Limit of a Function: if approaches as approaches (but ).

• Existence of Limit: The limit exists if both one-sided limits are equal.

Examples Using Graphs and Tables

• Piecewise Functions: Limits may differ from the function value at a point.

• Example: For , is undefined, but .

Section 2.3 – Techniques for Computing Limits

Limit Laws and Theorems

Several laws simplify the computation of limits for sums, differences, products, quotients, powers, and roots.

• Sum Law:

• Difference Law:

• Product Law:

• Quotient Law: (if denominator limit is not zero)

• Power Law:

• Root Law: (if for even )

Limits of Polynomial and Rational Functions

• Polynomial:

• Rational: (if )

Squeeze Theorem

The Squeeze Theorem helps find limits when a function is bounded between two others with the same limit.

• Theorem: If near and , then .

• Example: by bounding between and .

• 

Trigonometric Limits

• Example:

• Indeterminate Forms: Direct substitution may fail; use identities or algebraic manipulation.

Section 2.4 and 2.5 – Limits Involving Infinity

Infinite Limits and Limits at Infinity

Limits involving infinity occur when function values grow without bound or when the input variable approaches infinity.

• Infinite Limit: or if grows arbitrarily large as approaches .

• 

• One-Sided Infinite Limits: or ; or $-\infty$.

• 

• Limit at Infinity: if approaches a finite value as grows large.

• 

• Infinite Limits at Infinity: or if grows without bound as increases.

• 

Vertical and Horizontal Asymptotes

• Vertical Asymptote (VA): Occurs where the denominator of a rational function is zero and the function grows without bound.

• Horizontal Asymptote (HA): Occurs when approaches a finite value as approaches infinity.

• Degree Comparison:

  • If degree of numerator > degree of denominator: No HA

  • If degrees are equal: HA is ratio of leading coefficients

  • If degree of numerator < degree of denominator: HA is

End Behavior of Exponential and Logarithmic Functions

• Theorem 2.8: End behavior for , , and :

Section 2.6 – Continuity

Definition and Checklist for Continuity

A function is continuous at a point if its graph has no holes, jumps, or breaks. Most familiar functions are continuous everywhere, but some have points of discontinuity.

• Continuity Checklist:

  1. is defined (a is in the domain).

  2. exists.

• Point of Discontinuity: Where the function fails to be continuous.

Types of Functions and Continuity

• Polynomial Functions: Continuous everywhere.

• Rational Functions: Continuous where denominator is not zero.

• Composite Functions: If is continuous at and is continuous at , then is continuous at $a$.

• Root Functions: If is odd, is continuous wherever is continuous. If $n$ is even, $\sqrt[n]{f(x)}$ is continuous where .

• Inverse Functions: If is continuous and invertible on an interval, is also continuous on the image of that interval.

Continuity at Endpoints and on Intervals

• Right-Continuous:

• Left-Continuous:

• Continuous on Interval: Continuous at every point in the interval; at endpoints, continuous from the appropriate side.

Intermediate Value Theorem (IVT)

The IVT states that if is continuous on and is between and , then there exists in such that . This theorem guarantees the existence of a value but not its exact location.

• Application: Used to show that a solution exists for equations involving continuous functions.

• Example: If you invest $1000 after 5 years, IVT guarantees an interest rate between and will achieve this.