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Indeterminates
We Calculating have implicitly made use of the following theorem, which is an almost immediate consequence of the preliminary denition of a limit. Theorem 1. If f (x) = g(x) for x = c and either f or g has a limit at c, then both must have a limit at c and their limits must be the same. In eect, this is a reiteration of the fact that the value of a function at a limit point has no eect on a limit.
Typical Calculation
We use this theorem in the calculation of the limit almost every time we are faced with an indeterminate. The calculation generally looks like the following: limxc f (x) = = limxc g(x) = L. The idea is that we start with an indeterminate f (x) and keep simplifying, each time getting either a new form of f (x) or another function that is equal to f (x) except at c, until we have found some function g(x) whose limit is easy to nd.
Example
For example, the calculation in the previous example might look like the following: x2 + x 12 (x 3)(x + 4) limx3 = limx3 = limx3 (x + 4) = 7. x3 x3 To repeat, in a nutshell, we usually wind up calculating limits of indeterminates by simplifying until weve found a function, equal to the original except at the limit point, which is not indeterminate.
Types of Manipulations
Most of the time, the algebraic manipulation we need to do to simplify is in one of the following three categories. (1) Factor and cancel. (2) Rationalize and cancel. (3) Simplifying a complex fractional expression The example we saw fell in the rst category, factor and cancel. Consider the following examples in the other two categories. limx9 = limx9 x3 x3 x+3 = limx9 x9 x9 x+3 x9 1 1 = limx9 = . 6 (x 9)( x + 3) x+3
1
Example Rationalize and Cancel
2
Note: The algebraic manipulation used here was essentially the same as the technique of Rationalizing the Denominator often taught in elementary algebra, but we actually wind up rationalizing the numerator!
Simplifying a Complex Fractional Expression
Here, complex is used in the sense of complicated, where we have a rational expression (an algebraic expression that may be written or rewritten as a quotient of polynomials) where either the numerator or denominator is itself an expression involving fractions. Consider limz4 1 4 . The simplication here may be approached in z4 at least two dierent ways. Both are illustrated.
1 z
Technique 1
z 4z 4z 4z = limz4 = limz4 4z = limz4 limz4 z4 z4 z4 4z 1 1 1 = limz4 = . z4 4z 16 Here, we looked at the numerator as a dierence of fractions and calculated that dierence by rst getting a common denominator and then subtracting the numerators. We then looked at what was left as a quotient and calculated the quotient by inverting the denominator and multiplying. 1 z 1 4 4 4z
Technique 2
limz4
1 4 4z 4z 1 = limz4 = limz4 = limz4 = z4 z 4 4z 4z(z 4) 4z 1 z 1 4 1 z
1 . 16 Here, we observe the complications arose from the denominators of 4 4z . and z in the numerator and simply multiplied by 1 in the form 4z There will be other manipulations that will come up, and sometimes the simplication involved using these techniques may not be quite as straightforward, but most of the examples we run across will fall into one of these three cases.

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