Limits and Continuity

How to think precisely about approaching behavior, evaluate basic limits, and test whether a function is continuous at a point or over an interval.
Modified

April 26, 2026

Keywords

limits, continuity, one-sided limits, discontinuity, calculus

1 Role

This page is the entry point to single-variable calculus.

Its job is to make approach precise enough that derivatives, continuity arguments, and local approximation stop feeling like tricks.

2 First-Pass Promise

Read this page first in the single-variable calculus module.

If you stop here, you should still understand:

  • why a limit is about nearby behavior, not just the value at the point
  • how one-sided limits control piecewise and boundary behavior
  • how to test continuity at a point
  • what the common discontinuity types actually mean

3 Why It Matters

Calculus starts with a simple but subtle question:

what happens to f(x) when x gets close to a value?

That question sits under almost everything that comes later.

  • derivatives are defined as limits of difference quotients
  • continuity is the minimal promise that small input changes do not create arbitrary output jumps
  • optimization methods assume objectives change in interpretable local ways
  • numerical approximations depend on whether local behavior is smooth, singular, or unstable

If limits stay fuzzy, later calculus turns into rule memorization. If limits become clear, many later formulas start to feel inevitable.

4 Prerequisite Recall

  • a function assigns outputs to inputs in its domain
  • the graph, formula, and domain all matter
  • algebraic simplification can change how a formula looks without changing nearby behavior away from excluded points

5 Intuition

A limit asks about the behavior of the outputs as the inputs move close to a target value.

The crucial idea is:

the limit cares about what happens near the point, not necessarily at the point itself.

So these are different questions:

  • what is \(f(a)\)
  • what is \(\lim_{x \to a} f(x)\)

They may agree. They may disagree. One may exist while the other does not.

That difference is exactly why holes, jumps, and vertical blow-ups matter so much in calculus.

Continuity is the happy case where nothing pathological happens: the function is defined at the point, the limit exists, and the two match.

6 Formal Core

Definition 1 (Limit At A Point) We write

\[ \lim_{x \to a} f(x) = L \]

to mean that when \(x\) gets arbitrarily close to \(a\) from nearby values in the domain, the outputs \(f(x)\) get arbitrarily close to \(L\).

This statement is about nearby behavior. It does not require that \(x=a\) be used during the approach.

Definition 2 (One-Sided Limits) The left-hand limit

\[ \lim_{x \to a^-} f(x) \]

tracks approach from values smaller than \(a\).

The right-hand limit

\[ \lim_{x \to a^+} f(x) \]

tracks approach from values larger than \(a\).

A two-sided limit exists exactly when both one-sided limits exist and are equal.

Definition 3 (Continuity At A Point) A function \(f\) is continuous at \(a\) when all three conditions hold:

  1. \(f(a)\) is defined
  2. \(\lim_{x \to a} f(x)\) exists
  3. \(\lim_{x \to a} f(x) = f(a)\)

So continuity is the statement that the limiting behavior and the actual function value agree.

Proposition 1 (Common Continuity Rules) For first-pass calculus, the most important facts are:

  • polynomials are continuous everywhere
  • rational functions are continuous wherever the denominator is nonzero
  • sums, products, and compositions of continuous functions stay continuous where defined

These rules let you evaluate many limits by direct substitution, but only when continuity is already justified.

7 Worked Example

Consider

\[ f(x) = \begin{cases} \dfrac{x^2-1}{x-1}, & x \ne 1 \\ 3, & x = 1 \end{cases} \]

For \(x \ne 1\), the expression simplifies to

\[ \frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1. \]

So near \(x=1\), the graph behaves like the line \(x+1\), even though the original formula is undefined there before the piecewise repair.

Therefore

\[ \lim_{x \to 1} f(x) = \lim_{x \to 1} (x+1) = 2. \]

But the actual function value is

\[ f(1)=3. \]

So:

  • the limit exists
  • the function is defined at \(1\)
  • but the limit does not equal the function value

Hence \(f\) is not continuous at \(x=1\).

This is a removable discontinuity. If we redefine the point value to be \(f(1)=2\), the nearby behavior and point value would match, and the function would become continuous there.

8 Computation Lens

A practical first-pass workflow for limits and continuity is:

  1. check whether direct substitution is valid by continuity
  2. if substitution gives an indeterminate-looking form such as 0/0, simplify algebraically if possible
  3. if the function is piecewise or has a boundary, compute left-hand and right-hand limits separately
  4. once the limit is known, compare it to the actual function value if the question is about continuity
  5. classify any failure as removable, jump, or infinite when possible

This workflow is simple, but it prevents many early calculus mistakes.

9 Application Lens

In later optimization and ML pages, people often talk about local change, gradient information, or smooth objectives. Limits and continuity are the first layer beneath all of that.

  • if continuity fails badly, local approximation can break
  • if one-sided behavior disagrees, optimization near a boundary becomes more subtle
  • if a function blows up, numerical methods may become unstable

Even before multivariable calculus appears, this page teaches the habit of asking:

what does the function do near the point, and is that behavior compatible with the model I want to use next?

10 Stop Here For First Pass

If you can now explain:

  • the difference between \(f(a)\) and \(\lim_{x \to a} f(x)\)
  • why equal one-sided limits are required for a two-sided limit
  • the three-part continuity test
  • the meaning of removable, jump, and infinite discontinuities

then this page has done its main job.

11 Go Deeper

The strongest next steps after this page are:

  1. Derivatives and Local Approximation, because derivatives are built directly from limit ideas
  2. Optimization, if you want to see where continuity and local behavior become solver assumptions
  3. Backpropagation and Computation Graphs, if you want a later ML-facing reason the chain-rule side of calculus matters

12 Optional Deeper Reading

13 Optional After First Pass

If you want more practice before moving on:

  • sketch a graph with a hole and decide whether the limit exists there
  • build a piecewise function with unequal one-sided limits and classify the discontinuity
  • take a rational function and determine where direct substitution is valid versus where simplification is needed

14 Common Mistakes

  • confusing the point value with the limit
  • using direct substitution when continuity has not been justified
  • forgetting to compare left-hand and right-hand behavior for piecewise functions
  • saying a limit does not exist just because the function is undefined at the point
  • treating every discontinuity as the same kind of failure

15 Sources and Further Reading

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