Calculus

Limits: Sneaking Up on Answers

Sometimes you cannot just plug a number into a function and get an answer. Maybe the function has a hole, or it blows up, or it does something weird at one specific point. That is where limits come in.

A limit asks: what value does f(x) get close to as x gets close to some number a?

You do not care what happens at a — only what happens as you sneak up on it.

1. Approaching a Point

Consider the function f(x) = (x^2 - 1)/(x - 1). If you try plugging in x = 1, you get 0/0 — undefined. But what if you plug in values near 1?

f(x) = \frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1} = x + 1 \quad (x \neq 1)

The function simplifies to x + 1 everywhere except at x = 1, where there is a hole. Use the slider to approach x = 1 from both sides.

x (approach point)0.5

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\text{As } x \to 0.5, \quad f(x) \to 0.5 + 1 = \text{ about } 0.5 + 1

Try This

Try this: Drag x toward 1 from the left (values like 0.9, 0.99, 0.999) and from the right (1.1, 1.01, 1.001). The function value keeps approaching 2. Even though f(1) is undefined, the limit as x approaches 1 is 2.

The graph looks like the line y = x + 1, but with an invisible hole at x = 1. The limit “fills in” that hole by telling you what would be there.

2. Removable vs Jump Discontinuities

Not all holes are the same. A removable discontinuity is a single missing point that you could “plug” back in. A jump discontinuity is where the function leaps from one value to a completely different one.

Let’s compare. Use the slider to explore a function with a jump:

x (explore the jump)-0.5

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Connection

At x = 0, the left side gives f(x) = 0 + 1 = 1, but the right side gives f(x) = 2(0) - 1 = -1. Since the left and right limits disagree, the overall limit does not exist at x = 0. This is a jump discontinuity — no amount of “filling in” can fix it.

3. Left-Hand and Right-Hand Limits

Every limit is really two limits in disguise: the left-hand limit (approaching from smaller values) and the right-hand limit (approaching from larger values).

\lim_{x \to a^-} f(x) \quad \text0 \quad \lim_{x \to a^+} f(x)

The full limit exists only when both one-sided limits agree. Let’s visualize another function where this matters: f(x) = 1/x.

x (approach zero)0.5

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Try This

Try this: Approach x = 0 from the right (positive side) — the function shoots toward positive infinity. Now approach from the left (negative side) — it plunges toward negative infinity. The two sides completely disagree, so the limit at x = 0 does not exist. This is an infinite discontinuity.

4. When Limits Work Perfectly: Continuity

A function is continuous at a point if three things happen:

f(a) is defined (no hole)
The limit as x approaches a exists (left and right agree)
The limit equals f(a) (no surprise jumps)

\lim_{x \to a} f(x) = f(a) \implies \text{continuous at } a

Here is a smooth, continuous function — sin(x). Pick any point and the limit always matches the function value perfectly.

a (check continuity)2

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\lim_{x \to 2} \sin(x) = \sin(2)

5. Limits at Infinity

Limits also describe what happens as x goes to infinity. Does the function settle down to some value, or does it keep growing?

x (go toward infinity)5

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f(x) = \frac{2x + 1}{x + 3}, \quad f(5) \approx \frac{2 \cdot 5 + 1}{5 + 3}

Connection

As x gets enormous, the “+1” and “+3” become irrelevant compared to 2x and x. So the fraction behaves like 2x/x = 2. The horizontal line y = 2 is called a horizontal asymptote, and the limit at infinity is 2.

Challenge

Challenge: What is the limit of (3x^2 + x)/(x^2 - 4) as x goes to infinity? Hint: divide numerator and denominator by x^2, then see what happens to the leftover fractions as x gets huge.

The Big Idea

A limit tells you where a function is headed, even if it never actually arrives.

Limits are the language that makes calculus precise. Without them, we could not define derivatives (which are limits of slopes) or integrals (which are limits of sums). Every big idea in calculus is built on this one concept: sneaking up on an answer by getting closer and closer.

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