Limits, continuity and asymptotes

Adding a half, a quarter and successive halves of each part never produces exactly one in finitely many steps. Why is its limit 1? A limit describes an approximation that can be made arbitrarily accurate; equality at an individual step is unnecessary. Prerequisites: hyperbola, graph transformations, geometric sums and radians.

$S_n=\frac12+\frac14+\cdots+\frac1{2^n}=1-2^{-n},\qquad\lim_{n\to\infty}S_n=1.$

The remainder is 2⁻ⁿ. For any prescribed positive error, a step can be chosen after which every remainder is smaller. After step ten it is below 0.001. “Every later term”, rather than a few good approximations, is what convergence requires. Infinity means the index increases without bound; it is not a number to substitute.

Approaching a point is different from evaluating there

$g(x)=\frac{x^2-1}{x-1}=x+1\quad(x\ne1),\qquad\lim_{x\to1}g(x)=2.$

At x=1 the denominator vanishes, so g(1) is undefined. For every nearby permitted x, however, the graph agrees with x+1 and approaches 2 from both sides. The limit definition concerns x≠a.

Continuity

Hole at x=1

Jump at x=1

Same uniform scales, hypothetical dimensionless values. Filled points are included; open points excluded. Top: x+1; middle: same line without x=1; bottom: 1 to the left, 3 to the right including x=1.

The first panel f(x)=x+1 is continuous: its value and two-sided limit exist and agree:

$\lim_{x\to a}f(x)=f(a).$

The second lacks a value; filling its hole with 2 repairs it. The third hypothetical tariff is 1 for x<1 and 3 for x≥1. Left and right limits are 1 and 3, so no common limit exists: assigning one value cannot repair the jump. OpenStax

What an asymptote means

$\lim_{x\to0^+}\frac1x=+\infty,\quad\lim_{x\to0^-}\frac1x=-\infty;\quad\lim_{x\to\pm\infty}\frac1x=0.$

For 1/x, x=0 is a vertical asymptote and y=0 a horizontal one. The two sides have different signs; infinity is not a finite limit. A general graph can cross a horizontal asymptote: its definition describes behavior far away.

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When limit operations are valid

When both finite limits exist, the limit of a sum, difference or product is the corresponding operation on them. A quotient requires a nonzero denominator limit. A bounded monotone sequence converges; boundedness alone is insufficient, as (−1)ⁿ oscillates. Conditions follow the limit laws and monotone convergence theorem; general proofs are omitted here.

The fundamental trigonometric limit is:

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

The argument is in radians. Comparing triangle and sector areas in the unit circle gives cos x≤sin x/x≤1 for 0<x<π/2. Both bounds approach 1 and squeeze the expression to 1; evenness extends the result to negative x. With a degree argument the result would be π/180. Trigonometric limits.

Cauchy precision and continuity theorems

Assume domain points x≠a lie arbitrarily close to a. In the Cauchy definition, for every error ε>0 find a distance δ>0 such that all admissible x satisfy:

$0<|x-a|<\delta\quad\Rightarrow\quad|f(x)-L|<\varepsilon.$

Our g(x)=x+1 for x≠1 has |g(x)−2|=|x−1|, so δ=ε works. The distance depends on the requested precision, not on x chosen afterward. Precise definition.

The intermediate value theorem (Bolzano–Cauchy) says a continuous function on [a,b] takes every value between f(a) and f(b). Opposite endpoint signs therefore guarantee an interior zero. It guarantees existence, not uniqueness; without continuity the conclusion fails. Intermediate values.

Weierstrass’s theorems say continuity on a closed, bounded [a,b] guarantees boundedness and attainment of a maximum and minimum. For f(x)=x on open (0,1), boundary values 0 and 1 are never attained. The interval conditions matter; general proofs are omitted here. Extreme values.

Distinguish where values approach, what value is assigned at the point, and whether these agree. A hole and a jump give different answers even though both break continuity.