Limits and Derivatives

2.1 The Tangent and Velocity Problems

The study of calculus begins with two fundamental problems: finding the tangent to a curve at a point and determining the instantaneous velocity of a moving object. Both problems lead to the concept of the derivative.

• Tangent Problem: Involves finding the slope of the tangent line to a curve at a specific point. This is equivalent to finding the instantaneous rate of change of the function at that point.

• Velocity Problem: Concerns finding the instantaneous velocity of an object, which is the rate of change of its position with respect to time.

• Connection: Both problems are solved using limits, which measure how a function behaves as its input approaches a particular value.

• Example: The slope of the tangent to the curve y = f(x) at x = a is given by the limit:

2.2 The Limit of a Function

The concept of a limit is foundational in calculus. It describes the value that a function approaches as the input approaches a certain point.

• Definition: The limit of f(x) as x approaches a is L if f(x) gets arbitrarily close to L as x approaches a from either side.

• Notation:

• One-Sided Limits: Limits can be taken from the left () or right ().

• Example:

2.3 Calculating Limits Using the Limit Laws

Limit laws provide rules for evaluating limits of functions, making calculations systematic and reliable.

• Sum Law:

• Product Law:

• Quotient Law: , provided

• Example:

2.4 The Precise Definition of a Limit

The formal (epsilon-delta) definition of a limit provides mathematical rigor to the concept of limits.

• Definition: means that for every , there exists a such that if , then .

• Purpose: Ensures that the function values can be made as close as desired to L by taking x sufficiently close to a.

• Example: Prove using the epsilon-delta definition.

2.5 Continuity

A function is continuous at a point if its limit at that point equals its value there. Continuity is essential for many calculus theorems and applications.

• Definition: f is continuous at a if .

• Types of Discontinuity: Removable, jump, and infinite discontinuities.

• Example: The function f(x) = x^2 is continuous everywhere.

2.6 Limits at Infinity; Horizontal Asymptotes

Limits at infinity describe the behavior of functions as x grows without bound. Horizontal asymptotes are lines that the graph of a function approaches as x approaches infinity or negative infinity.

• Definition: means f(x) approaches L as x increases without bound.

• Horizontal Asymptote: The line y = L is a horizontal asymptote if or .

• Example: ; so y = 0 is a horizontal asymptote.

2.7 Derivatives and Rates of Change

The derivative measures the instantaneous rate of change of a function. It is a central concept in calculus, with applications in physics, engineering, and economics.

• Definition: The derivative of f at a is

• Interpretation: Represents the slope of the tangent line to the graph of f at x = a.

• Example: For f(x) = x^2,

2.8 The Derivative as a Function

The derivative can be viewed as a new function derived from the original, assigning to each input the instantaneous rate of change at that point.

• Definition: The derivative function f' is defined by

• Notation: , ,

• Example: If f(x) = x^3, then