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Lesson 1: The Intuitive Idea of a Limit

Estimated time: 30-40 minutes

Learning Objectives

By the end of this lesson, you will be able to:

Explain in your own words what a limit means
Estimate limits from a table of values
Estimate limits from a graph
Evaluate one-sided limits (left-hand and right-hand)
Determine when a two-sided limit does not exist

What Is a Limit?

Calculus is built on one powerful idea: the limit. Before we can talk about derivatives or integrals, we need to understand what happens to a function as its input approaches a particular value.

Imagine walking toward a door. A limit asks: "What position are you heading toward?" — not necessarily where you end up, but where you are approaching.

Limit (Informal Definition)

We write lim_x→a f(x) = L and say "the limit of f(x) as x approaches a equals L" if f(x) gets arbitrarily close to L as x gets arbitrarily close to a (from both sides), without necessarily requiring that f(a) = L.

The key phrase is "approaches but does not equal." When we compute lim_x→3 f(x), we look at what f(x) does when x is near 3 but not equal to 3. The function does not even need to be defined at x = 3 for the limit to exist.

Estimating Limits from a Table

One of the simplest ways to investigate a limit is to build a table of values. Choose x-values that approach the target from both sides and observe where f(x) appears to be heading.

Example 1: A Simple Polynomial Limit

Estimate lim_x→2 (x² + 1).

Step 1: Choose x-values approaching 2 from both sides.

x	f(x) = x² + 1
1.9	4.61
1.99	4.9601
1.999	4.996001
2	5
2.001	5.004001
2.01	5.0401
2.1	5.41

Step 2: Observe the pattern. As x approaches 2 from both sides, f(x) approaches 5.

Conclusion: lim_x→2 (x² + 1) = 5.

Example 2: A Limit Where Direct Substitution Fails

Estimate lim_x→0 (sin x)/x.

Step 1: Note that f(0) = 0/0, which is undefined. But the limit may still exist!

x	(sin x)/x
−0.1	0.99833
−0.01	0.99998
−0.001	0.9999998
0.001	0.9999998
0.01	0.99998
0.1	0.99833

Step 2: From both sides, (sin x)/x is heading toward 1.

Conclusion: lim_x→0 (sin x)/x = 1. This is one of the most important limits in calculus.

Estimating Limits from a Graph

Graphs provide a visual way to read limits. To find lim_x→a f(x), look at the height the curve approaches as x moves toward a, ignoring any dot or hole at x = a itself.

Example 3: Reading a Limit from a Graph

Suppose f(x) has a hole at x = 3 (open circle at (3, 4)) and a filled dot at (3, 2).

What is lim_x→3 f(x)?

Solution: Trace the curve from the left toward x = 3: the curve heads to height 4. From the right, the curve also heads to height 4. Therefore:

lim_x→3 f(x) = 4

Note: f(3) = 2 (the filled dot), but the limit is 4. The limit and the function value can differ!

Key Insight

The limit of f(x) as x → a depends on the behavior of f near a, not on the value of f at a. The limit can exist even if f(a) is undefined or equals a different number.

One-Sided Limits

Sometimes a function behaves differently from the left and from the right. We can describe this with one-sided limits.

One-Sided Limits

Left-hand limit: lim_x→a⁻ f(x) = L means f(x) → L as x approaches a from the left (x < a).

Right-hand limit: lim_x→a⁺ f(x) = L means f(x) → L as x approaches a from the right (x > a).

Example 4: Piecewise Function

Let f(x) = { x + 1 if x < 2, and 2x − 1 if x ≥ 2 }.

Find the one-sided limits at x = 2.

Left-hand limit: As x → 2⁻, use the rule f(x) = x + 1:

lim_x→2⁻ (x + 1) = 2 + 1 = 3

Right-hand limit: As x → 2⁺, use the rule f(x) = 2x − 1:

lim_x→2⁺ (2x − 1) = 2(2) − 1 = 3

Since both one-sided limits equal 3, the two-sided limit exists: lim_x→2 f(x) = 3.

When Does a Limit Not Exist?

A two-sided limit does not exist (DNE) in three common situations:

Case 1: One-Sided Limits Disagree

If lim_x→a⁻ f(x) ≠ lim_x→a⁺ f(x), then the two-sided limit does not exist. This happens at a jump discontinuity.

Case 2: Unbounded Behavior

If f(x) → ∞ or f(x) → −∞ as x → a, we say the limit does not exist (though we often write lim = ∞ to describe the behavior). Example: lim_x→0 1/x² = +∞.

Case 3: Oscillation

If f(x) oscillates infinitely often without settling on a value, the limit does not exist. Example: lim_x→0 sin(1/x) does not exist because it oscillates between −1 and 1.

Example 5: Jump Discontinuity

Let g(x) = { 1 if x < 0, and −1 if x ≥ 0 }.

lim_x→0⁻ g(x) = 1 and lim_x→0⁺ g(x) = −1.

Since 1 ≠ −1, lim_x→0 g(x) does not exist.

The Connection Between Limits and Function Values

There are three possible relationships between lim_x→a f(x) and f(a):

The limit equals the function value: lim_x→a f(x) = f(a). This is the "nice" case (the function is continuous at a).
The limit exists but differs from f(a): There is a "hole" in the graph at x = a, but the curve approaches a specific height.
The limit does not exist: The function jumps, blows up, or oscillates near x = a.

Example 6: Limit Exists but Function Undefined

Consider f(x) = (x² − 4)/(x − 2).

Step 1: f(2) is undefined because the denominator is zero.

Step 2: Factor the numerator: (x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2.

Step 3: Therefore lim_x→2 f(x) = lim_x→2 (x + 2) = 4.

The limit is 4 even though f(2) is undefined. The graph has a hole at (2, 4).

Infinite Limits (Preview)

Sometimes as x approaches a, the function values grow without bound. We describe this using the notation lim_x→a f(x) = ∞ or lim_x→a f(x) = −∞.

Example 7: Infinite Limit

Investigate lim_x→0⁺ 1/x.

x	1/x
0.1	10
0.01	100
0.001	1000
0.0001	10000

As x → 0⁺, 1/x grows without bound. We write: lim_x→0⁺ 1/x = +∞.

Similarly, lim_x→0⁻ 1/x = −∞.

Since the one-sided limits differ, lim_x→0 1/x does not exist.

Key Takeaways

A limit describes the value a function approaches as the input approaches a target, regardless of the actual function value at that point.
You can estimate limits using tables (plugging in nearby values) or graphs (reading the height the curve approaches).
One-sided limits examine behavior from the left (x → a⁻) or right (x → a⁺) only.
The two-sided limit exists if and only if both one-sided limits exist and are equal.
A limit does not exist if the one-sided limits disagree, the function is unbounded, or the function oscillates.
lim_x→0 (sin x)/x = 1 is a fundamental limit you will use frequently.

Check Your Understanding

1. If f(3) = 7 but the graph of f has a hole at (3, 5), what is lim_x→3 f(x)?

Answer: lim_x→3 f(x) = 5. The limit depends on where the curve is heading (the hole at height 5), not on the actual function value f(3) = 7.

2. Let h(x) = { 2x if x < 1, and x² + 2 if x ≥ 1 }. Does lim_x→1 h(x) exist?

Answer: Left-hand limit: lim_x→1⁻ 2x = 2. Right-hand limit: lim_x→1⁺ (x² + 2) = 1 + 2 = 3. Since 2 ≠ 3, the limit does not exist.

3. Use a table to estimate lim_x→1 (x² − 1)/(x − 1).

Answer: At x = 0.9: (0.81 − 1)/(0.9 − 1) = (−0.19)/(−0.1) = 1.9. At x = 0.99: 1.99. At x = 1.01: 2.01. At x = 1.1: 2.1. The values approach 2. You can verify by factoring: (x² − 1)/(x − 1) = (x+1)(x−1)/(x−1) = x + 1, so the limit is 1 + 1 = 2.

4. True or false: If f(a) is undefined, then lim_x→a f(x) does not exist.

Answer: False. The limit can exist even when f(a) is undefined. For example, lim_x→2 (x² − 4)/(x − 2) = 4, yet f(2) is undefined. The limit only cares about behavior near a, not at a.

Ready for More?

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In Lesson 2, you will learn algebraic techniques for computing limits exactly: direct substitution, factoring, rationalization, and limit laws.

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