Limits

Why limits exist (motivation)

When you evaluate a function f(x), you plug in a number x and get an output. That works well when the function behaves nicely everywhere. But many important questions are about behavior near a point, not at the point:

• •What height does a curve approach as x gets close to 2?
• •What slope does a curve approach as you zoom in (derivatives)?
• •What constant does a sequence of averages approach as sample size grows (law of large numbers)?
• •How does runtime grow as input size grows (Big O)?

The common pattern: we care about what happens as the input approaches something.

The intuitive definition

We write

lim_{x → a} f(x) = L

and read it as:

“As x approaches a, f(x) approaches L.”

This means the outputs f(x) can be made arbitrarily close to L by choosing x sufficiently close to a.

Two key ideas are hidden in this sentence:

1) x approaches a does not mean x = a. In fact, x might never equal a.

2) The limit depends on f(x) values near a, not on what happens at a.

“Approach” as a distance idea

Because you know the distance formula, it helps to translate “approaches” into distance:

• •“x approaches a” means the distance |x − a| becomes very small.
• •“f(x) approaches L” means the distance |f(x) − L| becomes very small.

So limits connect two distances:

• •small |x − a| ⇒ small |f(x) − L|

What limits are not

Limits are often confused with:

• •Substitution: f(a) is just plugging in x = a.
• •Value at a point: f(a) could be undefined or different from the limit.
• •A guarantee about far away behavior: a limit only describes behavior near a.

A quick picture in words

Imagine a function with a hole at x = 2, but the curve around it sits near y = 5. You might have f(2) undefined (a hole), yet

lim_{x → 2} f(x) = 5.

That is not a contradiction: limits ignore the single point x = 2 and care about points arbitrarily close to 2.

The formal (ε–δ) meaning, gently

You don’t need full formalism yet, but the core idea is worth stating:

lim_{x → a} f(x) = L means:

For every ε > 0, there exists δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε.

Interpretation:

• •ε controls how close you want f(x) to be to L.
• •δ tells you how close you must take x to a to achieve that.
• •The condition 0 < |x − a| excludes x = a, reinforcing that the limit doesn’t depend on f(a).

Even if you never write ε–δ proofs, this definition explains the “arbitrarily close” nature of limits.

Why evaluation methods matter

A limit is a behavioral claim. But in practice, you want to compute limits efficiently. The big strategy is:

1) Try direct substitution.

2) If substitution fails (often because of 0/0), simplify the function without changing its behavior near a.

3) If it still fails, use alternative viewpoints (graphs, tables, special limit facts, or more advanced tools later).

Method A: Direct substitution (when it works)

If f is “well-behaved” at a (no division by 0, no discontinuity), then the limit usually equals the function value:

lim_{x → a} f(x) = f(a).

Example pattern:

• •If f(x) = x² + 3x, then lim_{x → 2} f(x) = 2² + 3·2 = 10.

This is not the definition of a limit; it’s a consequence for continuous functions (a concept you’ll unlock soon).

Method B: The indeterminate form 0/0 (a sign to simplify)

A common situation:

lim_{x → a} \frac{g(x)}{h(x)}

where g(a) = 0 and h(a) = 0. Direct substitution gives 0/0, which is indeterminate: it does not mean the limit is 0, undefined, or anything by itself.

The point is: near x = a, the expression might simplify.

Classic tool: factoring and canceling

Suppose

f(x) = \frac{x² − 4}{x − 2}.

At x = 2, both numerator and denominator are 0. But:

x² − 4 = (x − 2)(x + 2)

So for x ≠ 2,

\frac{x² − 4}{x − 2} = \frac{(x − 2)(x + 2)}{x − 2} = x + 2.

Now the behavior near 2 is the same as the simpler function x + 2 (except at the single point x = 2).

So:

lim_{x → 2} \frac{x² − 4}{x − 2}

= lim_{x → 2} (x + 2)

= 4.

Notice what happened: we never needed f(2). In fact, the original expression is undefined at x = 2, yet the limit exists.

Method C: Rationalizing (when roots cause 0/0)

If you see something like:

\frac{√(x + c) − √(a + c)}{x − a}

direct substitution often yields 0/0. A standard move is to multiply by the conjugate.

Example structure:

(√u − √v)(√u + √v) = u − v.

That difference of squares is what removes the radical.

“Limit depends on near values, not the point” (a concept check)

Consider a function:

f(x) =

• •3x + 1, for x ≠ 2
• •100, for x = 2

Then:

lim_{x → 2} f(x) = lim_{x → 2} (3x + 1) = 7.

Even though f(2) = 100, the limit is 7 because all x values near 2 (but not equal to 2) use 3x + 1.

This is one of the most important limit intuitions: a single point does not control a limit.

A quick comparison table of evaluation approaches

Why one-sided limits are necessary

Sometimes “approach a” is ambiguous because the function behaves differently from the left and from the right. To capture that, we define:

• •Left-hand limit: lim_{x → a⁻} f(x)
• •Right-hand limit: lim_{x → a⁺} f(x)

The two-sided limit exists only if both one-sided limits exist and are equal:

If lim_{x → a⁻} f(x) = L and lim_{x → a⁺} f(x) = L,

then lim_{x → a} f(x) = L.

If they differ, the limit does not exist.

Example: a jump discontinuity

Define:

f(x) =

• •0, for x < 0
• •1, for x ≥ 0

Then:

lim_{x → 0⁻} f(x) = 0

lim_{x → 0⁺} f(x) = 1

Since 0 ≠ 1,

lim_{x → 0} f(x) does not exist.

This is not about being “undefined”; f(0) exists (it’s 1). The limit fails because there is no single output value the function approaches from both sides.

Infinite limits (blowing up)

Some functions grow without bound as x approaches a point. We express this with ±∞:

lim_{x → a} f(x) = ∞

This does not mean the limit is a real number. It means f(x) becomes arbitrarily large.

Example:

f(x) = 1/x².

As x → 0, 1/x² → ∞ from both sides (because x² is always positive).

So:

lim_{x → 0} 1/x² = ∞.

But with f(x) = 1/x:

lim_{x → 0⁻} 1/x = −∞

lim_{x → 0⁺} 1/x = ∞

The one-sided behaviors disagree, so:

lim_{x → 0} 1/x does not exist.

Limits at boundaries and “approaching infinity”

Sometimes you approach a boundary point of the domain (like x → 0⁺ for √x), or you study end behavior:

lim_{x → ∞} f(x), lim_{x → −∞} f(x).

These are still limits: “x grows without bound” is another form of “input approaches a target” (the target is not a finite number).

Example:

lim_{x → ∞} \frac{1}{x} = 0.

Interpretation: you can make 1/x as close to 0 as you want by taking x sufficiently large.

A practical checklist for “does the limit exist?”

When asked about lim_{x → a} f(x):

1) Check if both sides are relevant (is the function defined near a on both sides?).

2) Compute lim_{x → a⁻} f(x) and lim_{x → a⁺} f(x).

3) If they match to a finite L, the limit is L.

4) If they both go to ∞ (or both to −∞), describe it as an infinite limit.

5) If they disagree, the limit does not exist.

This checklist prevents a common error: assuming “a limit exists unless something is undefined.”

Derivatives: slope from “secant” to “tangent”

A derivative is defined using a limit. The slope of the secant line between x = a and x = a + h is:

m_secant = \frac{f(a + h) − f(a)}{h}.

The instantaneous slope (tangent slope) is what this approaches as h → 0:

f′(a) = lim_{h → 0} \frac{f(a + h) − f(a)}{h}.

Notice the same theme: we can’t just plug in h = 0 because that gives 0/0. Limits tell us what the expression approaches.

Continuity: “no breaks” is really “limit equals value”

A function is continuous at a (informally) if the graph doesn’t tear there. Formally, one key condition is:

lim_{x → a} f(x) = f(a).

So continuity is not separate from limits; it’s built from them.

Big O notation: limits as asymptotic comparison

In algorithm analysis, we care about growth as n → ∞. Limits formalize “dominates” comparisons. A classic comparison is:

lim_{n → ∞} \frac{n}{n²} = lim_{n → ∞} \frac{1}{n} = 0.

Interpretation: n grows much more slowly than n²; in asymptotic terms, n is negligible compared to n².

Even if Big O has its own formal definition, limit intuition is a huge help for understanding it.

Law of Large Numbers: convergence is a limit idea

In probability, you’ll meet statements like:

As n → ∞, the sample mean \bar{X}_n approaches the expected value μ.

That is a limit/convergence claim: a sequence of random quantities gets arbitrarily close to μ with high probability (formal versions use probability language, but the “approach” idea is the same).

Unifying idea: local vs global information

Limits are how math turns “zooming in” (local behavior near a point) into reliable statements. Many advanced concepts are just refinements of this:

• •derivatives (local slope)
• •integrals (limit of sums)
• •series (limit of partial sums)
• •asymptotics (limit comparisons)

If you understand limits as controlled approach, you’ll recognize the same pattern everywhere.