2.1 The Idea of Limits

Introduction to Limits

Limits are fundamental in calculus, providing a way to analyze instantaneous velocity and the slope of a tangent line to a curve. They allow us to rigorously define concepts that involve approaching a value, rather than reaching it directly.

• Limit: The value that a function (or sequence) approaches as the input (or index) approaches some value.

• Instantaneous Velocity: The velocity of an object at a specific instant, found as the limit of average velocities over shorter and shorter time intervals.

• Slope of the Tangent Line: The limit of the slopes of secant lines as the points get infinitely close.

Example: The position of a rock launched vertically can be modeled by . The average velocity over an interval is . The instantaneous velocity at is the limit as of .

2.2 Definitions of Limits

Numerical and Graphical Approaches

Limits can be estimated using tables of values and graphs. The value a function approaches as gets close to a certain point (from both sides) is the limit at that point.

• Numerical Estimation: Use a table of values for near the point of interest and observe the trend in .

• Graphical Estimation: Observe the behavior of the graph as approaches the value from the left and right.

Example: For as , the function is undefined at , but the limit can be estimated numerically and graphically.

One-Sided Limits

Sometimes, it is useful to consider the limit from only one side (left or right). These are called one-sided limits and are denoted as (from the left) and (from the right).

• If both one-sided limits exist and are equal, the two-sided limit exists and equals that value.

• If they are not equal, the two-sided limit does not exist.

2.3 Techniques for Computing Limits

Limit Laws

Several algebraic rules, called Limit Laws, simplify the computation of limits:

• Sum Law:

• Product Law:

• Quotient Law: , provided

Example: If , , and , then .

Limits of Polynomial and Rational Functions

If is a polynomial or rational function and is defined at , then .

• For rational functions, ensure the denominator is not zero at .

The Squeeze Theorem

The Squeeze Theorem is used when a function is "squeezed" between two others that have the same limit at a point. If for all near and , then .

2.4 Infinite Limits

Vertical Asymptotes

An infinite limit occurs when the values of a function increase or decrease without bound as approaches a certain value. This is often associated with vertical asymptotes.

• Example:

Limits at Infinity

Limits at infinity describe the behavior of a function as becomes very large or very small. This is used to find horizontal asymptotes.

• Example:

2.5 Continuity

Definition of Continuity

A function is continuous at if:

• is defined

• exists

Discontinuities can be classified as removable, jump, or infinite (vertical asymptote).

Continuity of Composite and Transcendental Functions

Composite functions are continuous at if is continuous at and is continuous at . Transcendental functions (exponential, logarithmic, trigonometric) are continuous on their domains.

2.7 Precise Definitions of Limits

Epsilon-Delta Definition

The formal (epsilon-delta) definition of a limit states: if for every , there exists a such that whenever , .

3.1 Introducing the Derivative

Definition and Interpretation

The derivative of a function at a point measures the instantaneous rate of change or the slope of the tangent line at that point. It is defined as:

Example: For , .

Tangent Lines and Rates of Change

Secant and Tangent Lines

The slope of the secant line between two points on a curve approximates the average rate of change. As the points get closer, the secant line approaches the tangent line, whose slope is the instantaneous rate of change (the derivative).

Summary List of Trig Identities

Additional info: These trigonometric identities are essential for simplifying expressions and solving problems involving derivatives and integrals of trigonometric functions.