Chapter 2: Understanding Limits and Rates of Change

Introduction

This chapter introduces the foundational concepts of calculus: limits and derivatives. You'll learn how these ideas help us solve two classic problems that seem impossible at first: finding the slope of a tangent line to a curve (the tangent problem) and finding the instantaneous velocity of a moving object (the velocity problem). Both problems lead to the same mathematical tool—the limit—which is the heartbeat of calculus.

2.1 The Tangent and Velocity Problems

The Tangent Problem

Imagine trying to draw a line that just barely touches a curve at a single point, following the curve's direction precisely at that spot. This is the tangent line to a curve. For circles, the definition is simple: a tangent is a line that touches the circle at exactly one point. But for more complicated curves, this doesn't work—a line can "touch" a curve at one point while actually crossing it at another.

Tangent Line: A line that touches a curve at a specific point and has the same direction (slope) as the curve at that point.

The clever approach is to use secant lines (lines that cross the curve at two points) to approximate the tangent. Here's the idea:

1. Pick a point on the curve where you want the tangent line
2. Choose a nearby point on the curve
3. Calculate the slope of the line through and (the secant line)
4. Move closer and closer to
5. The slopes of these secant lines get closer to the slope of the tangent line

Example: For the parabola at point :

• When is at , the secant slope is
• When is at , the secant slope is
• When is at , the secant slope is

As approaches , the secant slopes approach 2, so the tangent slope is 2, and the tangent line is , or .

This limiting process is the foundation for what we'll call the derivative later.

📝 Section Recap: The tangent line problem asks: how do we find the slope of a line touching a curve at a single point? By using secant lines through increasingly close points and taking their limiting slope, we can define the tangent line precisely—this limit process is essential to calculus.

The Velocity Problem

Now consider a different scenario: a car's speedometer shows different speeds as you drive through traffic. The car has an instantaneous velocity at each moment, even though we normally think of velocity as "distance over time" (which requires a time interval). How do we define instantaneous velocity when there's no time interval?

Suppose a ball falls freely from a height, and its distance fallen (in meters) after seconds is given by:

To find the velocity at exactly seconds:

1. Calculate the average velocity over a short time interval:

2. For the interval from to :

3. Now repeat with shorter intervals:

   • From to : average velocity m/s
   • From to : average velocity m/s
   • From to : average velocity m/s

As the time interval shrinks toward zero, the average velocities approach 49 m/s. This limiting value is the instantaneous velocity.

Instantaneous Velocity: The limit of average velocities over shorter and shorter time intervals. It represents the velocity at a single moment in time.

Notice something powerful: the tangent problem and velocity problem are mathematically identical! The slope of the secant line on a distance-time graph equals the average velocity. The slope of the tangent line equals the instantaneous velocity. Both require computing a limit.

📝 Section Recap: The velocity problem asks: how do we find the speed at a single instant when we only know how to calculate average speed over intervals? By taking average velocities over increasingly small time intervals and finding their limit, we can define instantaneous velocity—which geometrically equals the slope of the tangent line to the position graph.

2.2 The Limit of a Function

Understanding Limits Intuitively

Now that we've seen why we need limits, let's study them rigorously. The intuitive idea is straightforward:

Intuitive Definition of a Limit: We write (read: "the limit of as approaches equals ") if we can make the values of as close to as we want by taking sufficiently close to (but not equal to ).

Key insight: The value doesn't matter. We only care about the behavior of near . The function might not even be defined at —what matters is what happens in the neighborhood around .

Example: Consider . Wait—this simplifies to everywhere except at where it's undefined. But:

Even though doesn't exist, the limit exists because as approaches 1 from either side, approaches 1.

Another example: Let for near 1. Building a table:

And approaching from the right:

The values cluster around 0.5, so .

One-Sided Limits

Sometimes a function approaches different values from the left and right. Consider the Heaviside function, used in electrical engineering to model a switch that turns on:

As approaches 0:

• From the left, approaches 0
• From the right, approaches 1

These are one-sided limits:

One-Sided Limit (Left): means the values of approach as approaches from the left (with ).

One-Sided Limit (Right): means the values of approach as approaches from the right (with ).

For the Heaviside function: and .

Critical relationship: A two-sided limit exists if and only if both one-sided limits exist and are equal:

If the left and right limits differ, the two-sided limit does not exist.

Common Limit Examples

Example 1: Estimate

Building a table (calculator values):

• : function
• : function
• : function

The limit appears to be .

Note: Be cautious with calculators for very small values. When is extremely small, the numerator is nearly 0, and rounding errors can make the calculator give garbage results.

Example 2: Estimate (where is in radians)

This is a famous limit in calculus:

• : function
• : function
• : function

The limit is exactly 1. This result will be proven later using geometry.

📝 Section Recap: A limit describes the value a function approaches as the input gets closer to a target value. One-sided limits distinguish approaching from the left versus right. A two-sided limit exists only when both one-sided limits exist and agree. Limits can be estimated numerically (using tables) and graphically (by zooming in on a curve).

2.3 Calculating Limits Using the Limit Laws

Instead of building tables and graphs for every limit, we can use algebraic rules. Just as derivatives and integrals have rules, limits have laws.

The Limit Laws

Suppose and , where and are real numbers. Then:

1. Sum Law:

2. Difference Law:

3. Constant Multiple Law: for any constant

4. Product Law:

5. Quotient Law: provided that

6. Power Law: If is a positive integer, then

7. Root Law: If is a positive integer (and when is even), then

Direct Substitution

For polynomial functions and rational functions (where the denominator is nonzero at the point), you can find limits by direct substitution:

This works because polynomials and rational functions are continuous at every point in their domain.

Example: Find

By direct substitution:

Factoring and Cancellation

When direct substitution produces the indeterminate form , try factoring:

Example: Find

Direct substitution gives , which is indeterminate. Factor:

Now:

Rationalizing the Numerator

Sometimes multiplying by a conjugate helps:

Example: Find

Multiply by the conjugate:

Now:

The Squeeze Theorem

If you have a function trapped between two others that approach the same limit, the trapped function must also approach that limit:

Squeeze Theorem: If near and , then .

Example: Using the Squeeze Theorem to prove

We know that for all . Multiplying by (for ):

Since and , the Squeeze Theorem tells us . The same argument works from the left.

📝 Section Recap: The limit laws allow us to break complex limits into simpler pieces. Direct substitution works for continuous functions. When substitution gives , we use algebraic techniques: factoring, rationalizing, or the Squeeze Theorem to find the limit.

2.4 The Precise Definition of a Limit

The intuitive definition of a limit served us well, but for rigorous mathematics, we need precision. In this section, we give the epsilon-delta definition, which is the formal foundation for all limit work.

The Epsilon-Delta Definition

Precise Definition of a Limit: We write if for every number , there exists a corresponding number such that:

In plain language: No matter how close you want to get to (specified by ), you can always find an interval around (specified by ) such that if is in that interval (but not equal to ), then is within of .

Interpreting Epsilon and Delta

• (epsilon) represents the "tolerance" for how close should be to . It's given to us as a challenge: "Make within this distance of ."
• (delta) is our response: "If you let be within this distance of , then I can guarantee is within of ."

Proving Limits with Epsilon-Delta

The formal definition is used to prove that limits exist and equal specific values.

Example: Prove that

We need to show: for any , there exists such that if , then .

Simplify

We want , which means

So, if we choose , then whenever , we have ✓

When Limits Don't Exist

The epsilon-delta definition can also be used to show that a limit does NOT exist. A limit fails to exist if there's some such that no matter what you choose, you can find an with but .

Example: does not exist (where sign gives -1 for negative, 0 for zero, and 1 for positive), because the left and right limits are different.

📝 Section Recap: The epsilon-delta definition is the rigorous, formal definition of a limit. It quantifies what we mean by " approaches ": for any tolerance , we can find a window around where stays within of . This definition is the foundation for proving limit properties and showing when limits fail to exist.

2.5 Continuity

A function is continuous at a point if there's no break, jump, or hole in its graph at that point. Formally:

Definition of Continuity

Continuity at a Point: A function is continuous at if:

1. is defined
2. exists

In other words, the function is defined at the point, has a limit there, and the limit equals the function's value.

Continuity on an Interval

A function is continuous on an interval if it's continuous at every point in that interval.

Types of Discontinuities

• Removable discontinuity (hole): The limit exists, but either the function is undefined or . Example: at has a hole.

• Jump discontinuity: Left and right limits exist but are different. Example: the Heaviside function at .

• Infinite discontinuity: The function approaches . Example: at .

• Oscillating discontinuity: The function oscillates wildly. Example: at .

The Intermediate Value Theorem

If is continuous on the closed interval and is any number between and , then there exists at least one point in where .

Practical insight: If a continuous function is negative at one point and positive at another, it must cross zero somewhere in between. This is why root-finding algorithms work.

Which Functions are Continuous?

• Polynomials are continuous everywhere
• Rational functions are continuous except where the denominator is zero
• Roots and exponentials are continuous on their domains
• Trigonometric functions (sine, cosine) are continuous everywhere; tangent has discontinuities at odd multiples of

If and are continuous, so are , , , and (where ).

📝 Section Recap: A function is continuous at a point if the limit exists and equals the function's value. Discontinuities come in several types: removable holes, jump discontinuities, infinite discontinuities, and oscillations. Polynomials and other elementary functions are continuous on their domains. The Intermediate Value Theorem guarantees that continuous functions hit all values between any two inputs.

2.6 Limits at Infinity

What happens to as grows very large (positive or negative)? These "limits at infinity" describe the long-term behavior of functions.

Infinite Limits

Infinite Limit: We write if the values of become arbitrarily large as approaches .

For example, because as approaches 0 from the right, grows without bound.

Similarly, means becomes arbitrarily large and negative.

Limits as x Approaches Infinity

Limit as : We write if the values of approach as becomes arbitrarily large.

Example: because as grows without bound, gets closer to 0.

Example:

To find this, divide numerator and denominator by the highest power of :

As , the fractions , , and all approach 0, so:

Horizontal and Vertical Asymptotes

• Vertical asymptote at : The function has an infinite limit as (often written ).
• Horizontal asymptote at : or .

A rational function has:

• Vertical asymptotes where the denominator is zero (and numerator is nonzero)
• Horizontal asymptotes determined by comparing degrees:
  • If degree of < degree of : horizontal asymptote at
  • If degree of = degree of : horizontal asymptote at
  • If degree of > degree of : no horizontal asymptote

📝 Section Recap: Infinite limits describe functions growing without bound near a point. Limits at infinity describe long-term behavior as becomes very large. Vertical asymptotes are lines where functions shoot to infinity. Horizontal asymptotes are values that functions approach as . These concepts help us understand function behavior at extreme values.

2.7 Derivatives

We've now built all the tools needed to return to the tangent and velocity problems. The derivative is the formal name for the instantaneous rate of change.

The Derivative at a Point

Recall that the slope of the secant line through and is:

As , this approaches the slope of the tangent line. This limit is called the derivative of at :

Derivative (Definition): The derivative of at , denoted , is: provided this limit exists.

Equivalent form using :

Interpreting the Derivative

The derivative represents:

• The slope of the tangent line to the curve at the point
• The instantaneous rate of change of at
• For a position function, the instantaneous velocity

The Derivative as a Function

Instead of just finding at a single point, we can define the derivative function :

This function gives the derivative at every point in the domain.

Example: Find the derivative of

So for the function .

When Derivatives Don't Exist

A derivative fails to exist at a point if:

1. The function is discontinuous there
2. There's a sharp corner or cusp (left and right derivatives don't match)
3. The tangent line is vertical

Example: is not differentiable at because the left tangent slope is -1 and the right tangent slope is +1.

Notation

Different notations for the derivative:

• or — Lagrange notation
• or — Leibniz notation
• — operator notation
• — dot notation (common in physics for time derivatives)