What is a Limit, Part Two?

There is one further thing we need to say about limits before we move on. Instead of making your input number get closer and closer to the number at which everything falls apart, you can also make your input number just get bigger and bigger. This can also help us answer the unanswerable, because you cannot really make your input number infinitely large (you would need an infinitely powerful calculator to work out the answer), but you can say what would happen if you did.

Lets look again at our original function:

\[x+1\]

Lets switch the sign again:

\[x-1\]

Lets add an \(x\) to it:

\[2x-1\]

And lets divide it by \(x\):

\[\frac{2x-1}{x}\]

This time, lets call the function \(g(x)\), so we distinguish it from the last one. . What is \(g(1)\)? Well that is two times one (two) less one (one) divided by one. One divided by one is one. Algebraically:

\[\frac{2\cdot 1 -1}{1}\]

\[\frac{2-1}{1}\]

\[\frac{1}{1}\]

\[1\]

What about \(g(10)\)? That is two times ten (twenty) less one (nineteen) divided by ten. Nineteen divided by ten is one and nine tenths. So:

\[\frac{2\cdot 10 -1}{10}\]

\[\frac{20-1}{10}\]

\[\frac{19}{10}\]

\[1.9\]

Multiplying the input number by 10 has not made a massive difference to the output - it just added on 0.9. The reason why is that while the input number gets larger it appears on both the top AND bottom of the function so it multiplies and divides. Each operation (multiplication and division) just about cancels the other out. So what if we go really big? Lets try \(g(10000)\):

\[\frac{2\cdot 10000 -1}{10000}\]

\[\frac{20000-1}{10000}\]

\[\frac{19999}{10000}\]

\[1.9999\]

So no matter how big the number you stick in, you double it and divide it by itself which nearly cancels one of the copies completely, and would cancel it BUT FOR the one that you take away on the top line. This also works for whatever number you stick in - it does not have to be 1, 10, 100, 1000 or so on, it is just that the results from other numbers look a bit messy. What we are seeing here is that the bigger and bigger the starting number you put in, the closer and closer the output is to two. In mathematical terms we describe this as:

\[\lim_{x \to \infty} \frac{2x-1}{x}=2\]

Which means that as \(x\) gets closer to infinity (\(\infty\)) the output of the function gets closer to two.