Why Do Different Solving Methods Yield Different Results for (sin(x)/x) = x?

[k_squared]

Lets say I were trying to solve for x whereas (sin(x)/x) = x. Graphically, it can be shown easily that the answer is zero.

http://www.wolframalpha.com/input/?i=sin(x)+=x

But when I type in (sin(x)/x) = 1 in, there is an infinitesimal, but nonzero, solution.

http://www.wolframalpha.com/input/?i=(sin(x)/x)=1

Uhh... Why?

I'd really appreciate any response...

 

[someperson05]

Computers have inherent limitations. The answer is in fact still zero.

When a computer uses a numerical method to approximate the solution to an equation such as the one you gave, it simply can not calculate the answer exactly. It turns out that different representations of functions can give different computational results, which you have just seen.
(Why are computers ultimately inaccurate? Consider an irrational number like pi, 3.1415926... etc. Pi never ends. Well computers are inherently discrete systems and so a computer can NEVER represent an infinite decimal like pi. A computer only has so many bits and can only represent rational numbers ultimately. The computer's number line has "gaps".
Couple this limitation of computers with the following fact. It turns out that there are way more irrationals than there are rationals on the real number line. So much more in fact that if you picked a number at random off of the real number line, the probability that you would pick a rational is 0. Not 0.0000000001. Flat out zero.)

Moral of the Story:
This is one of the reasons computers are not the end all be all. A good thing to know.

 

[Integral]

Yet another example of a number that a digital computer cannot represent exactly is .1. When converted to binary it becomes an infintly repeating decimal and must be rounded off.

 

[CRGreathouse]

Yet another example of a number that a digital computer cannot represent exactly is .1.

You mean, "a binary computer" not "a digital computer". There are decimal computers (in, e.g., many calculators) that do not have this issue. (And of course they can be emulated even on binary computers, as in the Decimal type in C#.)

 

[CellCoree]

There are two different approaches to solving equations, graphically and algebraically. Graphically, we can see that the intersection of the graphs of sin(x)/x and x is at x=0, which is the solution to the equation (sin(x)/x) = x. This is because at x=0, the value of sin(x)/x is also 0.

However, when we solve algebraically by setting (sin(x)/x) = 1, we are essentially multiplying both sides by x, which can introduce extraneous solutions. In this case, x=0 is a solution to the equation (sin(x)/x) = 1, but it is not the only solution. This is why we see an infinitesimal, but nonzero, solution when we use this method.

It is important to understand the limitations and potential errors of different solving methods and to use them appropriately depending on the situation. In this case, the graphical method gives us a clear and accurate solution, while the algebraic method may introduce additional solutions that need to be carefully examined.