A math question

I’m really ignorant, possessing no knowledge of mathematics outside of High School, but i was looking at some exponential growth and decay functions, and i saw that the graph, although almost parallel to the x-axis, is slightly off by a fraction of a fraction. The value of y gets smaller and smaller the farther x gets away from zero. However, no matter what number you plug into the equation y=ab^x the value of y never reaches zero, meaning that the graph never crosses the y axis.

But the axis is infinite as well as the graph of the line, therefore with infinite possibility, shouldn’t the lines intersect sooner or later?

i can’t stop thinking about it, but i don’t have enough knowledge of math to do all the rest of the thinking that i need. if anyone could help me out that’d be great.

I dont think it will ever cross. I mean I dont even have knowledge of the type of function or whatever its called that your using, but just thinking about it this is what I come up with. If from your observance the lines have never crossed when plugging in x, then I would assume its because its an exponent. When ever we square or cube something, you never obtain a negative #, and zero is the only number that will illicit an answer of zero. Therefore it wont cross the y axis because it cant be zero(unless ab=0, then it will be truly parallel with the x-axis), and it cant be negative. The larger your exponent becomes the closer the line will come to crossing, but it never will for the reasons outlines above. This is just my opinion.

This is calculus. Consider a function of the form y = 1/x or y = 1/(2^x). If you take the limit of y as x approaches infinity, then the function y gets to zero, and can’t be less than zero. Imagine making x sufficiently big so that y reaches zero–this is pretty intuitive.

I wouldn’t touch this subject with a ten-foot pole, probably because I cant lift a ten-foot pole from one end.

Well, nevermind… I will go ahead and touch it.

Imagine an ant trying to get home, if he travels half the distance one day, the next day he will travel half of THAT distance, then half of that etc… So if he keeps going a half of a half, he will never reach home.

The line can never cross the x axis if the numerator is always zero, no matter if the denominator is 1 or 378234. It also works vice versa.

oooooo, I’m in earth science and in probably a few months im goingt to learn about decay functions!!! (half lifes, ect.) Im takin this in 8th grade and I would also like some decay functions ahead of the time I learn i9t in school.

[contented edited by ILP]

To learn about limits, you first need to look at continuity- Dedekind’s cuts. Differentiation provides information about a point by looking at its relation to neighbouring points. Speed is a continuous function so inorder to go from 5mph to 10 mph you must pass through all speeds in between. If at a point, gradients are taken to in relation to points nearer and nearer to it on both sides. The value on both sides will get closer and closer to one number- the limit. That limit is then given as the instantaneous speed at our fixed point, say 7mph. If you were to deny this, then our traveller would have somehow achieved a greater speed than 7, from 5, without passing through it. It as if you were to wake up one morning finding yourself 5 inches taller, but deny that you passed through all heights in between.

Instantaneous speed is man made, speed requires a change in time, but it is very useful.

Adapted From: WordNet 2.0 Copyright 2003 by Princeton University. All rights reserved.

asymptote : A straight line that is the limiting value of a curve; can be considered as tangent at infinity; “the asymptote of the curve”

In your example, I think the y-value approaches zero asymptotically.

well, your equation goes towards 0 only if b is in the interval (-1,1)-0 (can’t find acolades on this italian keyboard). if b is 1 or -1, then y is a^x and if both are 1 or -1, y is either 1 or -1. and if both a and b are greater then 1 (or smaller than -1), it will go towards infinity or -infinity.

it will never be 0, because of the ant :slight_smile: numbers smaller than 1 get smaller when raised to power. however, no matter how small the number and how great the power, they will never be 0, becuase a multiplication is 0 only when one of the factors is 0.

it’s like sailing towards the sunset. you can say that you are sailing towards the sunset, you have the direction, but will never reach sunset.

if it helps, try ro imagine that you function goes upwards. y is greater and greater, the limit is infinity, but can’t say that it will eventually reach infinity, because infinity isn’t a “place” (like 0).

Abgrund said it all. And what he didn’t say, rvw said :smiley: . It’s an asymptote. These things are very common.

I understand why everybody is talking about limits and assymptotes, but, these don’t explain why it is possible for a function to be always decreasing yet never cross a certain lower boundary.
The only person who, in my opinion, had it right was ApocalypseOfWar.
All ordered fields, which are mathematical structures (examples are the rational numbers (fractions) and the real numbers (decimals)) have a property sometimes refered to as the density property. This guarantees that given any two elements a, b such that a < b there is always an element x such that a < x < b. The proof is simple: take x = (a + b)/2.
This is what makes it possible for a function to be decreasing but to keep on taking values inbetween the boundary below and any function value that came before.

It will eventually reach zero when in stretches beyond your ability calculate it further.

This is real basic Calculus. In your example, the line y=ab^x is approaching 0 asymptotically, but will never actually interesect.

Think of a geometric sequence having the first four terms 1, 1/2, 1/4, 1/8.
As you can see, since the numbers in the sequence are going down by a factor of 2 with each subsequent number, the sequence is converging towards 0, but will never actually reach it. The fifth term will be 1/16, the tenth term will be 1/512. All of those numbers are very small, and they will continute to get smaller ad infinitum, but it will never actually reach 0.

Let’s look at the function f(x)=1/X. The limit of f(x) as x approaches infinity will result in 1 divided by a very large number, which will result in a very small number, but never zero. Because the only time f(x) will equal zero is when the numerator is zero but in this case it is one. And when X=0 the function is undefined because you can’t divide a number by zero. Why? Well, if you have 10 apples, you can’t divide it zero ways because even if you keep all the apples to yourself, you’re dividing by 1 not zero. I mean if you plan on not sharing the apples or keeping the apples, why will you want ten apples. It just doesn’t make sense to have any apples. And therefore, f(x) or sometimes referred to y will approach zero (really really small numbers) as X gets larger, but never actually reach zero.

Peace

I suppose another way of looking at it could be that powers of numbers are supposed to mean “multiply the number by itself X times” (so for any number N, N to the power of 2 means N times N, N to the power of 3 means N times N times N etc.)

Conveniently ignoring fractional and negative powers for the moment, this might help you see that for-any non-zero number N, it is impossible to multiply it by itself any number of times so that you get the answer zero. (because the only way to do that is to multiply the number by zero :slight_smile:)

Hmmm… Awaiting flames from mathmaticians…

I’m no mathmatician, but you’re absolutely correct. There is no way for an exponential function y=ab^n to equal zero (given that ab is not equal to zero).
When n is a postive number, y is a large number
When n equals zero, y equals 1
When n is a negative number (y=ab^-2 can be expressed as y=1/(ab^2). Hence you’ll have 1 divided by a large number which will be some small number but never zero.

Peace

Hmmm. Cheers. I think Troy McClure’s is the most appropriate answer yet, though. I think the original poster had problems envisaging how a line could get lower and lower without crossing a certain height, rather than being interested in the forumla itself.

But, yeah.

i’m tired and lazy, so i will nether read what others have allready posted nor can i rember what the actual term is, but i’m sure someone allready has or will mention the technicall term and reason for it. there are certain exrpesions on that result in a line that expodentialy grows closer and closer to zero by smaller and smaller intervals that divide over and over, but never actualy reach zero, though the line may go on forever. meh, i think the figures are usualy put in terms xpi or in root form because it’s an irrational number.

Well, it can never cross is the simple answer.
One of the most fundamental theorems of calculus is that of hte limit. A limit is essentially sendnig X to a certain number. For instance:
1/0 = undefined is what we are taught in math.
however, simply keep divinding by lower and lower numbers and you would find this answer out yourself:
1/.1 = 10
1/.01 = 100
1/.001 = 1,000
1/.0001= 10,000

As you can see, it is heading towards Oinfinity.

However, with a limit, you aren’t actaully dividing by the number. As above, you aren’t dividing by 0 in 1/0, but rather 1/.000000000000000000000001 and smaller numbers.
I’m not sure if this is exactly what you are lookng for, but the simple answer is no.
(In addition, assymptotes can be crossed in some graphs such as e^-x * sinx I believe. This graph, once I again, if i remember correctly, osscilates around the x axis but is a dampened function, it crosses it’s assymptote infinitie times, but it is none the less an assymptote._

–There is No Truth