🌋Infinite Limits & Asymptotes

Warden Zeno keeps the rim of Limit's Edge, where the ground stops and the value of a function does not. Everything on this rim comes from one situation: a fraction whose bottom is heading for zero while its top is heading for something else.

Think about 1/x as x shrinks. At x = 0.1 the value is 10. At x = 0.01 it is 100. At x = 0.000001 it is a million. Nothing stops it. The values do not approach a number, so the limit does not exist as a number — but saying only "does not exist" throws away the interesting half of the story, which is *which way* the values ran.

So we say it in words instead. The values grow without bound in the positive direction, or in the negative direction. That is a real description of behaviour, and it is what a vertical asymptote is: the line x = a that the graph climbs beside without ever touching.

How to test a candidate. Substitute x = a into the fraction.

• ◆Bottom zero, top not zero: a vertical asymptote. The graph runs away.
• ◆Bottom zero, top zero: the 0/0 form, which is not an answer at all. Factor and cancel first, then look again.
• ◆Bottom not zero: no asymptote. Substitution just works.

That first case is the one this lesson lives in, and once you have found it, only one question is left: which way does it run?

Finding the direction. Never guess it, and never reach for a calculator. Count signs, one factor at a time, just to the side you were asked about.

Take 5 / (x - 3) as x approaches 3 from the right. "From the right" means x is a hair bigger than 3 — think x = 3.001. Then x - 3 = 0.001, a tiny positive number. A positive 5 divided by a tiny positive number is a huge positive number. The values grow without bound in the positive direction.

Now approach 3 from the left, x = 2.999. Now x - 3 = -0.001, a tiny negative number, and 5 divided by that is a huge negative number. The same function, the same point, opposite directions.

That is why a two-sided limit at a simple asymptote does not exist in any usable sense: the two sides disagree. Squaring changes everything, though. In 5 / (x - 3)² the bracket is squared, so it is positive on both sides, and the values run positive from both directions.

A zero of the bottom is only a candidate. Whether it becomes an asymptote depends on what the top is doing at the same place, and the only way to find out is to factor both lines and look for a bracket they share.

Consider two functions that look almost identical.

f(x) = (x - 2) / (x² - 5x + 6)

g(x) = (x + 4) / (x² - 5x + 6)

Both bottoms factor as (x - 2)(x - 3), so both have the candidates x = 2 and x = 3.

For g, nothing cancels. The top, x + 4, is 6 at x = 2 and 7 at x = 3, neither of them zero. Both candidates are genuine vertical asymptotes.

For f, the bracket (x - 2) appears on both lines. It cancels, leaving

f(x) = 1 / (x - 3) for every x except 2

At x = 2 the function still has no value — you cannot divide 0 by 0 — but the graph does not run away there. It has a single missing point: a hole. Only x = 3 is a vertical asymptote.

The difference matters because it changes the picture completely. Beside an asymptote the graph shoots off the top or the bottom of the page. At a hole the graph carries on as if nothing happened, with one dot lifted out of it. On a screen you would never see the hole at all.

The rule, in one line. Cancel first, then read the asymptotes off what is left. A bracket that cancelled gives a hole at its zero; a bracket left underneath gives an asymptote at its zero.

One more case worth knowing: a bottom like x² + 4 has no real zero at all, because x² is never negative and so x² + 4 is at least 4 everywhere. A graph like that has no vertical asymptote anywhere.

A vertical asymptote asks what happens near one x. The opposite question asks what happens when x runs off to the far end of the graph, and the answer is written the same way: as a limit.

If f(x) settles onto a single height c as x grows without bound, the line y = c is a horizontal asymptote, and we write it as a limit of f(x) as x approaches infinity. The two ideas are the same fact seen twice.

For a quotient of polynomials, the whole answer is decided by comparing the two degrees, because out at the ends the highest power on each line dwarfs everything else.

Bottom degree bigger. The bottom outgrows the top, so the fraction is crushed towards zero and the horizontal asymptote is y = 0. For (3x + 5) / (2x² - 1), divide everything by x²: the top becomes 3/x + 5/x², which heads for 0, while the bottom heads for 2. The limit is 0.

Degrees equal. The two leading terms grow at the same rate, so the limit is the ratio of the leading coefficients — with their signs. For (6x² - x) / (4x² + 9), divide by x² and everything but 6/4 fades away, so the horizontal asymptote is y = 3/2. Notice it is the leading coefficients that matter, never the constants: those are the terms that disappear.

Top degree bigger. Nothing holds the fraction back and it runs away, exactly as a polynomial does. There is no horizontal asymptote, and the answer is again a direction rather than a number.

The direction at the far left. As x decreases without bound, work out the sign of the leading power first. An even power of a huge negative number is positive; an odd power is negative. So -2x³ runs positive at the far left and negative at the far right, while -2x⁴ runs negative at both ends.

One degree apart is special. When the top is exactly one degree ahead, long division splits the function into a straight line plus a remainder that fades away:

(x² + 5x + 3) / (x - 1) = x + 6 + 9/(x - 1)

There is no horizontal asymptote, but the graph presses against the slanting line y = x + 6. That is an oblique (or slant) asymptote.

Radicals follow the same plan — divide by the biggest thing in sight — with one extra rule that catches almost everybody the first time.

A square root is never negative. So for huge x, sqrt(x^2 + 1) behaves like the *size* of x, which is written |x|, and not like x itself.

• ◆At the far right, x is positive, so |x| is x.
• ◆At the far left, x is negative, so |x| is -x.

That single minus sign means a radical graph can approach two *different* horizontal lines, one at each end. Watch it happen with

f(x) = (3x + 1) / sqrt(x^2 + 4)

Divide the top and the bottom by |x|. The bottom becomes sqrt(1 + 4/x²), which heads for 1 at both ends. The top is where the ends part company: dividing 3x by x gives 3, but dividing 3x by -x gives -3. So the graph approaches y = 3 on the right and y = -3 on the left, and both answers are correct — for their own end.

Whenever a square root of a squared term appears, ask yourself which end you are at *before* you cancel.

Exponentials are simpler, and go the other way. A base larger than one keeps doubling as its exponent rises and keeps halving as its exponent falls, so:

• ◆e^x grows without bound as x grows, and heads for 0 as x decreases without bound
• ◆e^-x, which is the same as 1 divided by e^x, does exactly the reverse

A decaying term never quite reaches zero, which is what makes y = 0 an asymptote rather than a point on the graph. And when a constant is standing beside the decaying term, the constant is what survives:

the limit of 20 - 8e^-x as x grows without bound is 20

because the 8e^-x piece fades away and the 20 does not. The 8 rides on the piece that vanishes, so it vanishes with it. That is why a cooling or a charging model always flattens onto its constant.