Unit 1: Limits and Continuity

Limits are the language of calculus. They describe how a function behaves near a point, even when the function value at that point is missing, undefined, or unrelated to the nearby trend.

What a limit means

We write

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

if we can make \(f(x)\) as close to \(L\) as we want by taking \(x\) sufficiently close to \(a\), with \(x \ne a\).

This is about nearby behavior, not direct substitution. It is possible for:

  • the limit to exist while \(f(a)\) is undefined,
  • the limit to exist while \(f(a) \ne L\),
  • the limit to fail even though \(f(a)\) exists.

Limit laws

If \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), then:

\[\lim_{x \to a} (f(x) \pm g(x)) = L \pm M\] \[\lim_{x \to a} (f(x)g(x)) = LM\] \[\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, \qquad M \ne 0\] \[\lim_{x \to a} [f(x)]^n = L^n\]

For polynomials and rational functions, direct substitution works whenever the denominator is nonzero.

One-sided limits and existence

A two-sided limit exists exactly when both one-sided limits exist and agree:

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

If the left-hand and right-hand limits disagree, the limit does not exist.

Common reasons a limit fails to exist:

  • jump discontinuity,
  • vertical asymptote with unbounded behavior,
  • oscillation, such as \(\sin(1/x)\) near \(x = 0\).

[Image Placeholder: left-hand vs right-hand limit examples, including a jump discontinuity]

Squeeze Theorem

If

\[g(x) \le f(x) \le h(x)\]

for all \(x\) near \(a\), and

\[\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L,\]

then

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

For example:

\[-1 \le \sin(1/x) \le 1\]

so

\[-\lvert x \rvert \le x\sin(1/x) \le \lvert x \rvert\]

implies

\[\lim_{x \to 0} x\sin(1/x) = 0.\]

Continuity

A function is continuous at \(x = a\) if:

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

Types of discontinuities:

  • removable: a hole, often fixable by redefining one point,
  • jump: left and right limits differ,
  • infinite: vertical asymptote,
  • oscillatory: no single nearby trend.

Functions continuous on their natural domains:

  • polynomials,
  • rational functions where denominator is nonzero,
  • exponential and logarithmic functions on domain,
  • trig functions on domain,
  • compositions of continuous functions where defined.

Intermediate Value Theorem

If \(f\) is continuous on \([a,b]\) and \(N\) lies between \(f(a)\) and \(f(b)\), then there exists some \(c \in (a,b)\) such that \(f(c) = N\).

This theorem does not tell you how many such points there are, only that at least one exists.

[Image Placeholder: continuous curve crossing a horizontal line to illustrate IVT]

Limits at infinity and asymptotic behavior

We also study

\[\lim_{x \to \infty} f(x), \qquad \lim_{x \to -\infty} f(x).\]

For rational functions:

  • degree numerator < degree denominator: limit is \(0\),
  • equal degrees: limit is ratio of leading coefficients,
  • degree numerator > degree denominator: no finite horizontal asymptote.

Useful asymptotic idea:

\[\sqrt{x^2 + 1} \sim \lvert x \rvert\]

for large \(\lvert x \rvert\), but be careful with sign when \(x \to -\infty\).

Indeterminate forms

These forms require more work; they do not determine the limit by themselves:

  • \[0/0\]
  • \[\infty/\infty\]
  • \[0 \cdot \infty\]
  • \[\infty - \infty\]
  • \[1^\infty\]
  • \[0^0\]
  • \[\infty^0\]

At this stage, most are handled by algebra, conjugates, factoring, or trig identities. Later, some are handled with L’Hopital’s Rule.

Important trig limits

Two core limits:

\[\lim_{x \to 0} \frac{\sin x}{x} = 1\] \[\lim_{x \to 0} \frac{\tan x}{x} = 1\]

These only work cleanly in radians.

Average rate of change as a precursor to derivative

On \([a,b]\),

\[\frac{f(b) - f(a)}{b-a}\]

is the average rate of change, or slope of the secant line. The derivative will be the limit of this expression as the interval collapses.

Common algebraic techniques for limits

  • Factor and cancel removable factors.
  • Rationalize with conjugates when radicals are involved.
  • Rewrite complex fractions as a single fraction.
  • Use common denominators.
  • Extract dominant powers for end behavior.

Example:

\[\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} = \lim_{x \to 4} \frac{1}{\sqrt{x} + 2} = \frac14\]

BC-minded notes

  • Piecewise-defined functions often test whether you can make a function continuous or differentiable by solving for unknown constants.
  • A removable discontinuity may still break differentiability if not fixed.
  • Sequence limits later mirror function limits, except the input variable is restricted to integers.

Common mistakes

  • Plugging in before checking whether substitution is legal.
  • Claiming a limit does not exist just because the function is undefined at the point.
  • Forgetting that trig limit formulas require radians.
  • Treating vertical asymptotes as automatic DNE without checking one-sided behavior.
Made by Joshua Bie