Chapter 2: Limits and Continuity#

How do you know what a function is doing right near a point, even if the function never quite reaches that point? That question leads to the idea of a limit, the foundation of calculus. Once we understand limits, we can talk precisely about continuity, make sense of infinity, and solve problems where plain arithmetic fails.

The Big Picture#

Everything in calculus—derivatives, integrals, the very idea of smooth change—depends on the concept of a limit. A limit tells us what value a function is heading toward as the input gets close to some number. In this chapter we start with simple intuition, build a toolbox of limit laws and algebraic tricks, learn to handle infinity and tiny oscillations, and finally get a rigorous definition. Along the way we meet the squeeze theorem, special trigonometric limits, and continuity. By the end, you will be able to compute many limits, decide when a function is continuous, and use the Intermediate Value Theorem to find hidden roots. No panic—this is just about getting comfortable with “approaching.”

Intuitive Limits and One-Sided Limits#

Imagine you are walking straight toward a wall. Each step brings you closer; your distance to the wall gets smaller and smaller. You can get as close as you like, even if you never actually touch the wall. The limit of your distance, as you approach the wall, is zero. That is the flavor of a limit.

We write

and say, “the limit of as approaches is .” It means we can make as close to as we want, provided we take close enough to (but not equal to — the function’s actual value at , if there is one, does not matter).

Sometimes we only approach from one side. A one-sided limit restricts the approach. The left-hand limit,

looks at what happens when is a little less than (from the left). The right-hand limit,

looks at what happens when is a little greater than (from the right).

Here is a concrete example. Let

If we slide toward 0 from the left, stays at 1. If we slide in from the right, stays at 2. So

The two one-sided limits are different. Therefore the full two-sided limit does not exist.

Relationship between two-sided and one-sided limits:
if and only if both and .

In other words, the two-sided limit is the one number that both one-sided limits must agree on. That simple test will become second nature.

📝 Section Recap: A limit describes the value a function approaches as the input gets near a number. One-sided limits check from only the left or the right, and the full limit exists exactly when both one-sided limits are equal.

Limit Laws and Direct Substitution#

Computing limits one by one from scratch would be tedious. Fortunately, limits follow predictable arithmetic rules. If we know that and both exist, then:

These are the limit laws, and they make life much easier.

Many important functions have a wonderful property: if is in the domain, you can just plug it in. For polynomials, rational functions (where the denominator is nonzero), root functions (where defined), trigonometric, exponential, and logarithmic functions, we have direct substitution:

This is essentially what “continuous” means, which we will formalize later. For now, the practical message is: whenever you see one of those friendly functions and the point is in its domain, substitute directly.

Example:

The limit laws justify why this works, but the calculation is painless. The only hitch comes when direct substitution gives a meaningless expression like ; that is a signal to use algebraic tricks.

📝 Section Recap: Limit laws let us break complicated limits into simpler pieces. Direct substitution works whenever the function is “nice” and the point is in its domain—but we must be ready for indeterminate forms.

Algebraic Techniques for Indeterminate Forms#

Sometimes direct substitution yields , , or other indeterminate forms. These do not mean the limit fails; they mean we have to dig deeper. The function might have a removable hole that algebra can reveal.

Factoring and canceling

Consider . Putting gives . Factor the numerator:

For we can cancel the factor, leaving , which approaches as . So

Rationalizing

When square roots (or cube roots) appear, multiplying by a conjugate often clears up the trouble. Take . Substitution gives . Multiply numerator and denominator by :

Now let . The expression becomes .

The key is to rewrite the expression so the troublesome factor cancels. These techniques—factoring, expanding, rationalizing, using trigonometric identities—form the algebra toolkit for limits.

📝 Section Recap: When direct substitution gives an indeterminate form like , look for a hidden algebraic simplification. Factoring or rationalizing often turns a messy limit into a straightforward one.

Infinite Limits and Vertical Asymptotes#

Sometimes a function grows without bound as approaches a particular value. For example, near the function gets huge. We capture this behavior by saying the limit is infinite:

Here and are not real numbers; they signal unbounded growth in the positive or negative direction. If a one-sided limit is infinite, the graph has a vertical asymptote at that -value—a vertical line that the graph gets as close as we like to but never touches (or touches only at infinity).

A rational function can have a vertical asymptote where its denominator is zero but the numerator is not. For instance, has a vertical asymptote at , and both sides approach :

Infinite limit (informal): means becomes arbitrarily large and positive as gets near —the values go higher than any given number.

The sign matters. For , the limit from the right () is , and from the left is , because the denominator changes sign.

📝 Section Recap: Infinite limits describe unbounded behavior. A vertical asymptote occurs at when a one-sided limit equals or , and the sign tells us whether the graph shoots up or down.

Limits at Infinity and Horizontal Asymptotes#

Instead of letting approach a finite number, we can ask what happens when gets very large. A limit at infinity describes the long-run behavior of a function:

If such a finite limit exists, then (or ) is a horizontal asymptote—a horizontal line that the graph settles near as goes far to the right or left.

The simplest example is :

For rational functions, we compare the degrees of numerator and denominator. If , the limit is . If the degrees are equal, the limit is the ratio of leading coefficients. If the numerator’s degree is larger, the limit is infinite (no horizontal asymptote).

Example: Evaluate .

Both numerator and denominator have degree 2. Divide each term by :

As , the terms , , all go to , so the limit is . Thus is a horizontal asymptote.

📝 Section Recap: Limits at infinity reveal the eventual behavior of a function. A horizontal asymptote occurs when the function approaches a constant value as heads toward , and for rational functions this depends on the degree comparison.

The Squeeze Theorem and Special Trigonometric Limits#

Two very special limits show up continually in calculus:

We can prove these with the Squeeze Theorem (also called the Sandwich Theorem). The idea is delightfully simple. Suppose you have three functions, and one is always sandwiched between the other two:

for all near (except possibly at ). If the two outer functions have the same limit , then the middle function is forced to have that limit as well. The two “slices of bread” squeeze the filling to a definite value.

For , consider a small angle (in radians) inside the unit circle. A geometric argument shows that for ,

As , , so both outer bounds race toward . The squeeze theorem forces

A symmetric argument from the left () gives the same limit. Thus the two-sided limit is .

The cosine limit follows by a little algebra:

As , the first factor , and . So the product tends to :

These two limits help solve many harder limit problems, especially when arguments like appear: for example, after the substitution .

📝 Section Recap: The Squeeze Theorem sandwiches a function between two others that share a limit. It gives us the fundamental trigonometric limits and as , which we use again and again.

The ε-δ Definition of a Limit#

So far we have spoken about “getting close” in an intuitive way. The rigorous definition uses two tiny positive numbers: ε (epsilon) and δ (delta). It may look formal, but it simply makes our intuition solid.

ε-δ definition of a limit: We write if for every there exists a such that
if , then .

In plain words: tell me how close you want to get to (that tolerance is ), and I can find a narrow window around (the band) so that every input in that window (except possibly itself) gives an output within the tolerance.

Think of throwing a dart at a target. Suppose the bullseye is , and you want the dart to land within of it. If you can guarantee that by keeping your aim within a distance of , you have a limit.

Let’s prove a simple limit: . Given , we want

Choose . Then whenever , we get . The definition holds.

Developing good ε-δ skills takes practice, but the core idea is negotiation: for any challenge , you respond with a working .

📝 Section Recap: The ε-δ definition formalizes our intuitive idea of a limit. It says that outputs can be forced to lie arbitrarily close to by restricting inputs to be sufficiently close to . This precision underpins all limit laws and continuity proofs.

Continuity at a Point and on an Interval#

A function is continuous at a point if you can draw it through that point without lifting your pencil. More precisely, three things must be true:

Continuity at a point: is continuous at if

1. is defined,
2. exists,
3. and .

If any of these fails, there is a discontinuity. For example, a removable discontinuity (a “hole”) occurs when the limit exists but is not equal to the function value—or the value is missing entirely. A jump discontinuity happens when the one-sided limits exist but are different. An infinite discontinuity occurs when a one-sided limit is infinite.

A function can be continuous on an open interval if it is continuous at every point inside. If the interval includes endpoints, we use one-sided continuity at the edges.

The graph of a continuous function over a closed interval has no breaks, jumps, or holes—the pencil never leaves the paper.

Example of a hole: . For , this simplifies to , so , but is undefined. The graph has a hole at .

Example of a jump: the greatest integer function jumps at every integer. At , and , so the limit does not exist; the function is discontinuous.

📝 Section Recap: A function is continuous at a point when the limit and the function value agree. Discontinuities can be removable (holes), jump, or infinite. Over an interval, continuity means no breaks—an unbroken curve.

Building Continuous Functions#

Many common functions are continuous wherever they are defined. Specifically:

• Polynomials are continuous everywhere.
• Rational functions are continuous everywhere their denominators are nonzero.
• Root functions (with even) are continuous on their domain (); odd roots are continuous for all real .
• Trigonometric functions (, , etc.) are continuous everywhere they are defined; and have vertical asymptotes where .
• Exponential functions () and logarithmic functions are continuous on their natural domains (all reals for exponentials, for logs).

Moreover, continuity is preserved by the usual algebraic operations. If and are continuous at , then:

• , , ( any constant), and are continuous at .
• is continuous at provided .

Compositions also behave well. If is continuous at and is continuous at , then the composite function is continuous at . This means a function like is continuous everywhere because and are both continuous everywhere.

These facts explain why we can frequently rely on direct substitution: most functions we meet in calculus are continuous on their domains, so whenever belongs to the domain.

📝 Section Recap: Polynomials, rational, root, trigonometric, exponential, and logarithmic functions are continuous on their domains. Sums, products, quotients (where defined), and compositions of continuous functions are continuous, so limits for such functions often reduce to direct substitution.

The Intermediate Value Theorem#

A continuous function has a powerful property: it cannot skip values. Suppose you are driving along a road (your position is a continuous function of time). If at 2:00 you are at mile 10 and at 2:05 you are at mile 15, then sometime between 2:00 and 2:05 you must have passed through mile 12. The Intermediate Value Theorem (IVT) makes this intuition precise.

Intermediate Value Theorem: If is continuous on a closed interval and is any number between and (inclusive), then there exists at least one in such that .

One of the most useful consequences of the IVT is locating roots of equations. If and have opposite signs (one positive, one negative), then lies between them, so the equation must have at least one solution between and .

Example: Show that has a solution between and .

Define , a polynomial, hence continuous everywhere. Compute and . Since , the IVT guarantees a number in with . We may not know the exact root, but we know it exists and we can narrow it down by repeatedly bisecting the interval.

📝 Section Recap: The Intermediate Value Theorem says a continuous function on an interval takes every value between its endpoint values. This lets us prove that roots exist by checking for a sign change, a powerful tool even when the root itself is hard to find.

Summary#

Limits describe what value a function is heading toward. We learned to compute limits using limit laws, algebra, and special tricks. We handled infinite limits and limits at infinity, and we saw the squeeze theorem and the precise ε-δ definition. Continuity means the limit equals the function value, and the Intermediate Value Theorem helps us find roots. These ideas are the foundation for derivatives and integrals.