[FR] Indeterminate Forms

Indeterminate forms are algebraic expressions obtained when taking a limit. This most often occurs when the limit of dependent variable x approaches either 0 or ∞. However, that is not to say that algebraic expressions in which the dependent variable approaches 0 or ∞ are indeterminate forms.

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For example: the limit as x approaches ∞ of 1/x is 0.

You might be thinking, "How does dividing 1 by ∞ make 0?"

Consider the function 1/x and make x increase to ∞.

x = 1                 1/x = 1/1 = 1
x = 10               1/x = 1/10 = 0.1
x = 100             1/x = 1/100 = 0.01
x = 1000           1/x = 1/1000 = 0.0001
x = 10000         1/x = 1/10000 = 0.00001
...

Notice as we take x to be sufficiently large, the value of 1/x decreases to 0.
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Problems arise when limits of fractions form a type 0/0 or ∞/∞. While the unknowing mathematician would say a/a is 1, he forgets the facts that you cannot divide by 0 and ∞ is more a direction rather than a value or constant, which is why these are limits.

So why does 0/0 or ∞/∞ give us a problem?
Consider 0/0:
If the numerator approaches 0 faster (that is, at a greater rate) than the denominator, then the limit becomes 0 itself.

However, if the denominator approaches 0 faster than the numerator, then the limit becomes ∞.

Two opposite sides of the spectrum.

The concept is the same for ∞ in that if the numerator approaches faster, then the limit is ∞, and if the denominator approaches faster, than the limit is 0.