Everyone knows that it's impossible to divide a number by 0. Dividing
by zero just doesn't make sense on paper, and if you try it on a
calculator, you just get an error! Mathematically, we say that the
answer is undefined.
But if you could divide
by zero, what would the answer be?
Suppose you have an expression dependent on a variable, x, which you
represent as f(x). For example
suppose f(x) is defined as
follows, where I've added the condition that x must be greater
than 4:
f(x) = 1/(x −
4), x
> 4
In other words, f(x) is undefined for
values of x
of 4 or less. Even without this condition, x is undefined at
4, since that would make f(x) = 1/(4-4)
= 1/0, which is of course undefined. Here is a plot of the
function:
In this case, as x
approaches 4 from the right, we see that the value of f(x) gets
dramatically higher and higher, and the we get to x = 4,
the faster the function shoots up. Moving away from x = 4, we see that
the opposite is true: as x
increases, f(x) gets closer and
closer to zero, but never reaches it.
Think about it: divide any number by a very large number, and you get a
very very small number. Or divide by a negative number, and you get a
negative result; there's nothing you can divide by to get exactly zero.
Going the other way, if you divide by a fraction, you get a result
which is bigger than the original number; the smaller the fraction, the
larger the result - but how can you divide by zero?
We can, however, say what the results would be if you could. For example,
as x gets
larger and larger and approaches infinity
(infinity is a hypothetical
quantity, with the symbol ∞. It represents an impossibly
large number
- too large to actually exist. Indeed, that's what we need here if we
have any hope of getting f(x) to equal 0),
then f(x) will approach
zero. But you can never actually reach infinity - it is by definition
larger than the highest number you can ever reach. So, we say that the limit
of f(x) as x approaches
infinity, is zero. We can write it as follows:
lim
f(x) = 0 ^{x → ∞}
An alternative way of stating this is simply to write: "As
x → ∞, f(x) → 0."
Likewise, As x → 4, f(x) → ∞:
lim
f(x) = ∞ ^{x →
4}
So although f(x) can never
actually be equal
to infinity, the limit
of f(x) as x approaches 4 is infinity.