Limits, Continuity & Asymptotic Comparison

A limit describes behavior near a point without requiring anything special at the point itself. That separation is what makes derivatives possible at all: a difference quotient is undefined at zero step, yet its nearby values may approach one number. Continuity then adds the missing agreement between nearby behavior and the value at the point, while asymptotic notation compares the scale of errors and growth.

Why approaching a value is useful #

Some quantities are defined by an experiment that cannot be performed at the exact limiting case. Instantaneous speed would require dividing motion by a zero-length time interval; a limit instead studies ordinary nonzero intervals and asks whether their answers settle toward one value.

The same pattern appears whenever a finite resolution is refined: a numerical mesh, a sensor sampling interval, or an approximation parameter stays nonzero in every actual calculation, while the limit states what those calculations must approach.

A limit is a quantified contract #

The order of the quantifiers is the substance: an adversary may demand any positive output tolerance $\varepsilon$; only then may we choose a suitable input tolerance $\delta$. The choice may depend on $\varepsilon$, $a$, and the function, but not on the particular $x$ later selected inside the punctured neighborhood.

The condition excludes $x=a$. Therefore changing one function value does not change the limit:

$$f(x)=\frac{x^2-1}{x-1}\quad(x\ne1), \qquad f(1)=100$$

still satisfies

$$\lim_{x\to1}f(x) =\lim_{x\to1}(x+1)=2,$$

even though $f(1)=100$. The factor cancellation is legal only for $x\ne1$, which is exactly the region the limit examines.

For nonlinear expressions, first restrict $x$ to a simple neighborhood so that uncontrolled factors become bounded. To prove $\lim_{x\to1}x^2=1$, write

$$\lvert x^2-1\rvert=\lvert x-1\rvert\lvert x+1\rvert.$$

If we additionally require $\lvert x-1\rvert<1$, then $0<x<2$ and $\lvert x+1\rvert<3$. Thus $\delta=\min(1,\varepsilon/3)$ works. The min pattern is not decoration: one bound controls the geometry, the other delivers the requested error.

Direction and infinity matter #

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

$$\lim_{x\to a}f(x)=L \quad\Longleftrightarrow\quad \lim_{x\to a^-}f(x)=L=\lim_{x\to a^+}f(x).$$

For $f(x)=\lvert x\rvert/x$, the left-hand limit at zero is $-1$ and the right-hand limit is $1$, so the two-sided limit does not exist. Writing “the limit is $\pm1$” loses the crucial fact that there is no single output value satisfying the contract.

An infinite limit is a statement about unbounded magnitude, not a real-valued limit. For example,

$$\lim_{x\to2}\frac{1}{(x-2)^2}=+\infty$$

means that for every threshold $M>0$, values sufficiently close to but different from $2$ make the function exceed $M$. It does not mean that $+\infty$ is a number the function reaches. The vertical line $x=2$ is a vertical asymptote because at least one one-sided magnitude diverges there.

A limit at infinity instead moves the input:

$$\lim_{x\to+\infty} f(x)=L$$

means that for every $\varepsilon>0$ there is a threshold $N$ such that $x>N$ implies $\lvert f(x)-L\rvert<\varepsilon$. If this finite limit exists, $y=L$ is a horizontal asymptote as $x\to+\infty$. The limits at $+\infty$ and $-\infty$ are independent.

Computing limits without losing conditions #

Limit laws preserve sums, products, and quotients when the component limits exist; for a quotient, the denominator limit must be nonzero. Continuous elementary functions permit direct substitution wherever they are defined. When substitution produces an indeterminate form, algebra exposes the actual behavior.

“Indeterminate” means the form alone does not determine the answer. It never means the limit fails to exist.

The squeeze theorem is the right tool when the middle expression oscillates but its amplitude is controlled. Since

$$-x^2\le x^2\sin(1/x)\le x^2,$$

both outer functions tend to zero, so the middle function does too. A plot suggests this; the inequalities prove it.

L’Hôpital’s rule is conditional #

For a quotient $f/g$ approaching $0/0$ or $\pm\infty/\pm\infty$, L’Hôpital’s rule can replace the quotient by $f'/g'$ when the functions are differentiable on a punctured interval, $g'(x)\ne0$ there, and the derivative quotient has an appropriate limit (finite or infinite). Endpoint and one-sided variants use corresponding intervals.

$$\lim_{x\to a}\frac{f(x)}{g(x)} =\lim_{x\to a}\frac{f'(x)}{g'(x)}$$

is a theorem under those hypotheses, not a symbolic cancellation rule. Check the form before differentiating. For example,

$$\lim_{x\to0}\frac{\sin x}{x} =\lim_{x\to0}\frac{\cos x}{1}=1.$$

But applying the rule to $x/(1+x)$ at zero would be invalid because the original form is $0/1$, already determinate. Repeated use is allowed only after rechecking the hypotheses and the new form at every step.

Continuity connects the limit to the value #

This definition distinguishes common failure modes:

• a removable discontinuity has a finite limit but the value is absent or different;
• a jump discontinuity has finite but unequal one-sided limits;
• an infinite discontinuity has unbounded one-sided behavior;
• an oscillatory discontinuity, such as $\sin(1/x)$ at zero, has no limiting output despite remaining bounded.

Continuity is preserved under sums, products, quotients with nonzero denominator, and compositions whose intermediate value lies in the outer function’s domain. The domain clause prevents a familiar mistake: $\sqrt{g(x)}$ is not a real-valued continuous composition where nearby $g(x)$ becomes negative.

On a closed interval $[a,b]$, continuity upgrades local control to global guarantees:

• the extreme value theorem guarantees attained minimum and maximum values;
• the intermediate value theorem guarantees every value between $f(a)$ and $f(b)$ is attained.

The intermediate value theorem proves existence, not uniqueness and not a numerical approximation by itself. Bisection adds an algorithm and a certificate.

Uniqueness needs another argument, such as strict monotonicity. Here $f'(x)=3x^2-1>0$ on $[1,2]$, so the bracket contains exactly one root.

Asymptotic notation compares scales #

Limits answer “what value?” Asymptotic notation often answers “at what scale?” Following the NIST DLMF conventions, as $x\to c$:

$$f(x)\sim g(x) \quad\Longleftrightarrow\quad \frac{f(x)}{g(x)}\to1,$$

$$f(x)=o(g(x)) \quad\Longleftrightarrow\quad \frac{f(x)}{g(x)}\to0,$$

and

$$f(x)=O(g(x))$$

means $\lvert f(x)/g(x)\rvert$ remains bounded in a sufficiently small punctured neighborhood of $c$. The limiting process must always be stated or obvious from context.

These relations are not interchangeable:

• $\sin x\sim x$ as $x\to0$;
• $1-\cos x\sim x^2/2$ as $x\to0$;
• $\sin x-x=O(x^3)$ and, more sharply, $\sin x-x\sim-x^3/6$ as $x\to0$;
• $x^2=o(x)$ as $x\to0$, but $x=o(x^2)$ is false there.

Asymptotic equivalence is excellent inside products and quotients when denominators remain meaningful. It is dangerous under subtraction because leading terms may cancel. From $f\sim g$ one cannot conclude $f-g\sim0$—the expression “$\sim0$” is generally meaningless because the defining ratio would divide by zero. Instead measure the remainder with $o$ or $O$ notation.

Asymptotes are remainder statements #

A horizontal asymptote $y=L$ at $+\infty$ means $f(x)-L\to0$. A slant asymptote $y=mx+b$ means

$$f(x)-(mx+b)\to0$$

in the chosen direction. If $m$ and $b$ exist, they can be recovered as

$$m=\lim_{x\to\infty}\frac{f(x)}{x}, \qquad b=\lim_{x\to\infty}\bigl(f(x)-mx\bigr).$$

For

$$f(x)=\frac{x^2+1}{x-1},$$

polynomial division gives

$$f(x)=x+1+\frac{2}{x-1},$$

so $y=x+1$ is a slant asymptote at both $+\infty$ and $-\infty$. The same function also has a vertical asymptote at $x=1$; these facts describe different limiting processes.

A reliable diagnostic workflow #

1. State the process: $x\to a^+$, $x\to-\infty$, $n\to\infty$, or another explicit direction.
2. Check the real domain: roots, logarithms, inverse trigonometric functions, and denominators may restrict the approach.
3. Try continuity/direct substitution. Record the resulting value or indeterminate form.
4. Choose a transformation with reversible local algebra: factoring, a conjugate, a common denominator, a dominant scale, or a logarithm.
5. Use a theorem only after checking its hypotheses: squeeze inequalities, continuity, monotonicity, or L’Hôpital.
6. Preserve the remainder: if leading terms cancel, compute the next scale instead of reusing a coarse equivalence.
7. Sanity-check both sides and signs. A two-sided answer must agree with both one-sided behaviors.

The unifying question is always the same: what exact contract is being claimed, and which conditions make it true? Once that contract is explicit, most limit calculations become controlled transformations rather than pattern matching.