Foundations — Infinite limits and limits at infinity — vertical - horizontal asymptotes

Before you can read the parent note comfortably, you need every piece of its vocabulary built from nothing. This page is the toolbox. Each item below gives you plain words → the picture → why the topic needs it, and each one leans on the one before it.

1. A function $f(x)$ — the height machine

The picture: stand at position $x$ on the ground line, look straight up (or down) to the curve — that vertical distance is $f(x)$.

Why the topic needs it: asymptotes are statements about how the height behaves, so we need a clear name for "height at position $x$". That name is $f(x)$.

The symbol $y$ is just another name for the height: writing $y=f(x)$ means "let $y$ stand for the output". That is why the parent writes asymptotes as $y=L$ (a height) and $x=a$ (a position) — here $L$ is just a fixed number, a special height the curve heads toward; we meet it properly in section 7.

2. Domain — the legal inputs

The picture: the ground line with a few points punched out — those punched-out spots are inputs the function refuses.

Why the topic needs it: vertical walls can only appear at inputs the function is not defined on. When the parent divides by $(x-3)$, the value $x=3$ is thrown out of the domain, and that missing spot is exactly where a wall (or a hole) may live. So the domain tells you where to even look for vertical asymptotes.

3. The number line, $x\to a$, and "approaching"

The picture: a point walking toward $a$, first at distance $1$, then $0.1$, then $0.001$… squeezing in but never touching.

Why we forbid landing on $a$: often $a$ is not even in the domain — $f(a)$ is undefined (you divided by zero there). The whole game of limits is asking "what value is the height heading toward near $a$?" — a question that makes sense even when $f(a)$ doesn't exist.

4. Distance and absolute value $|x-a|$

The picture: a ruler laid between the points $x$ and $a$; $|x-a|$ is the length you read off, no matter which of the two is on the left.

Why the topic needs it: "$x$ gets close to $a$" is vague. "$|x-a|$ is small" is precise — it says the ruler-distance is tiny. This is the exact meaning of the $\delta$ (delta) in the formal definitions.

5. One-sided approach: $x\to a^{+}$ and $x\to a^{-}$

The picture: two separate dots — one sliding leftward into $a$ from higher ground, one sliding rightward into $a$ from lower ground.

Why the topic needs it: near a vertical wall the height can rocket up from one side and down from the other (exactly what $1/x$ does at $0$). The little $+$ and $-$ superscripts let us describe each side on its own. See One-sided limits.

6. The signs of tiny numbers: $0^{+}$ and $0^{-}$

Why the topic needs it: the entire "which direction does the wall go?" question is answered by tracking signs:

This is the machinery behind every vertical-asymptote example in the parent note.

7. Infinity $\infty$ — a direction, not a number

The picture: a ceiling you keep raising — $10$, then $10^6$, then $10^{100}$ — and the curve eventually pokes above each one. That "beats every ceiling" behaviour is what we call $+\infty$.

Why the topic needs it: both headline ideas involve $\infty$. Infinite limit = the height $\to \pm \infty$. Limit at infinity = the input $x\to \pm \infty$. Two different things wearing the same symbol — keep them apart.

8. The special height $L$ and the challenge numbers $M$, $N$, $\epsilon$, $\delta$

First, the height the curve settles toward:

The next four letters power the formal definitions. Think of limits as a two-player game: a challenger dares you, and you must always answer.

The picture: the challenger draws a thin horizontal band of half-width $\epsilon$ around the ceiling $y=L$; you must find how far right ($N$) the curve must go to slip inside that band and never leave.

Why the topic needs it: these make "close" and "big" into checkable promises instead of hand-waving. Full detail lives in Limits — formal epsilon-delta definition; here you only need to recognise the letters.

9. Powers, roots, and "highest power wins"

Why the topic needs it: comparing the top degree $m$ against the bottom degree $n$ (bigger, equal, smaller) is the whole rule for horizontal asymptotes of rational functions. That comparison is meaningless without the idea "highest power wins far out".

10. Polynomial and degree

The picture: the degree is the term that grows fastest — the tallest sky-scraper far out to the right, the one that decides the skyline.

Why the topic needs it: the parent's whole horizontal-asymptote rule is a size contest between $m$ and $n$ ($m<n$, $m=n$, $m>n$). Without "polynomial", "$P(x)$/$Q(x)$", and the degrees $m,n$, that rule has no words.

11. Rational function — a ratio of polynomials

The picture: two height-machines fighting — the top pulls the value up, the bottom pushes it down; wherever the bottom hits zero you get a possible wall, and far out the two leading terms decide the ceiling.

Why the topic needs it: these are the main worked examples of the parent. Their walls come from zeros of $Q$ (the punched-out domain points); their ceilings come from comparing degrees $m$ and $n$. (When $m$ is exactly one more than $n$, you get a tilted ceiling instead — see Slant (oblique) asymptotes via polynomial division.)

12. Removable hole vs genuine wall

The picture: a smooth curve with one pixel poked out, versus a curve that genuinely rockets to the sky.

Why the topic needs it: it stops the classic blunder "denominator $=0$ ⇒ vertical asymptote". Always inspect the numerator too. Deeper treatment: Continuity and removable discontinuities.

Prerequisite map

Read it top to bottom: the domain and one-sided approach feed the sign machinery that builds vertical walls; powers feed the polynomial degrees that build horizontal ceilings; the game pieces make both rigorous.

Equipment checklist

Test yourself — reveal only after answering aloud.

Connections

• Parent: Infinite limits & asymptotes
• Limits — formal epsilon-delta definition
• One-sided limits
• Continuity and removable discontinuities
• Curve sketching using first and second derivatives
• Slant (oblique) asymptotes via polynomial division
• L'Hôpital's rule for indeterminate forms