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Limits

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.

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