Introduction to Limits

Algebra asks what equals at a particular . Calculus asks a subtler question: as gets closer and closer to a number, what value is heading towards? That value is called the limit, and it’s the idea that every later topic in calculus is built on. Derivatives, integrals, continuity and infinite series are all limits underneath.

What Is a Limit?

Here is the definition of a limit in calculus, in plain words: means that gets as close to as you like when is close enough to , from either side. Three details matter:

• approaches but never has to equal it. The limit is about the neighbourhood of , not the point itself.
• Both sides count. Values of just below and just above must lead to the same .
• is a single, finite number. If the values run off to infinity or never settle, there is no limit.

You read the notation as “the limit of as approaches equals ”. The arrow is the important symbol: it means “gets closer and closer to”, never “equals”. This informal idea is all you need for now; the precise version, with and , comes at the end of this chapter.

Example 1: A Function With a Hole

Find .

You can’t just plug in : that gives , which isn’t a number. So doesn’t exist. But the limit doesn’t need . It only asks what happens near 1, so try some nearby values.

From both sides the outputs close in on 2. The graph shows why: for every , the expression simplifies to , because . So the graph is the line with a single point missing.

The function has no value at 1, yet the limit is perfectly clear. That gap between “the value at a point” and “the value it approaches” is exactly what limits are designed to handle.

Example 2: When You Can Just Substitute

Find .

Nothing goes wrong at , and the graph is a smooth curve with no breaks. As approaches 3, approaches :

For polynomials and most everyday functions at points where they’re defined, the limit equals the value. Those functions are called continuous, and they have a lesson of their own later in this chapter. The interesting limits are the ones where substitution fails, as in Example 1.

The Limit Doesn’t Care About f(a)

All three graphs follow the same curve towards the point . The first has , the second has a hole, and the third has moved the point to . In every case , because the limit only looks at the values around 2. The piecewise functions lesson shows how functions like the third one are built.

Example 3: Estimating a Limit With a Table

Estimate , with in radians.

Substituting gives again, and this time no algebra cancels the problem. A table gives a convincing estimate:

Because is even, negative inputs give exactly the same outputs. The values close in on 1, so . Tables are a good way to estimate, but they never prove anything: a function could change behaviour closer in than you checked. The squeeze theorem, later in this chapter, proves this one properly.

When a Limit Does Not Exist

A limit has to be one finite number that both sides agree on. It fails in three typical ways:

1. A jump. The left side heads for one value and the right side for another, so there’s no single . Comparing the two sides separately is the idea behind one-sided limits.
2. Unbounded growth. Near a vertical asymptote, grows without bound. We sometimes write to describe how it fails, but infinity isn’t a number, so the limit still doesn’t exist.
3. Oscillation. swings between and 1 faster and faster as and never settles on any value.

Why Limits Matter

Suppose a ball has fallen metres after seconds. What is its speed at exactly ? Speed is distance over time, but at a single instant no time passes, so you get . Instead, take the average rate of change over a short interval from 1 to :

For the average speed is 10.5, 10.05 and 10.005 m/s. As , it approaches 10 m/s, the instantaneous speed. That limit is a derivative, and it’s where Calculus 1 is heading.

Common Mistakes to Avoid

• Treating the limit as . They match for continuous functions, but the limit can exist when doesn’t, or differ from it.
• Stopping at . It doesn’t mean the limit is 0 or doesn’t exist. It means more work is needed: simplify, or estimate.
• Checking only one side. Both sides must approach the same value.
• Trusting a table as proof. A table suggests a limit; algebra or a theorem confirms it.

Practice Problems

Work each one before opening the answer.

Problem 1. Find .

Problem 2. Find .

Problem 3. Use a table to estimate .

Problem 4. A function has for , and . Find and .

Problem 5. Does exist?

Next, you’ll read limits straight off a graph, including graphs with holes, jumps and asymptotes.