Indeterminate form

Indeterminate Forms UHospital s Theorem Forms of the type 0/0, oo/oo, 0 X oo, etc., are callea indeterminates. To find the limiting values that the corresponding functions approach, L Hospital s theorem is useful If two functions/(x) andg(x) both become zero at X = a, then the hmit of their quotient is equ to the hmit of the quotient of their separate derivatives, if the limit exists or is -i- oq or — oo. 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

An alternative way to find a special formula for a special case is to apply L Hospital s rule to the general case. When bo ao. Equation (1.34) has an indeterminate form of the 0/0 type. Differentiating the numerator and denominator with respect to bo and then taking the limit gives... 

This introduces the gas-phase residence time VgjQg as a new parameter. It also introduces an ambiguity regarding the term kgAi(a — Ug). There is no resistance to mass transfer within a pure component so kgAj oo and a — Ug 0. Thus, kgAi(a — Ug) is an indeterminate form of the oo x 0 variety. Its value must continue to equal the rate at which oxygen is transferred into the liquid phase. Equation (11.5) remains true and the pair of simultaneous ODEs become... 

If two functions fix) and g(x) both vanish at a point a, the ratio f(a)/g(a) is undefined. It is the so-called indeterminate form mentioned earlier... 

L HospitaTs rule has been applie abpve to cases in which the indeterminate form is . However, it is equally valid for the form... 

This result, , is the most common indeterminate form (see Section 2.8). 

By applying L Hopital s rale for indeterminate forms, show that equation 3.4-11 results from... 

An analytical proof of the first of these limiting cases follows directly from equations 15.3-3 and -4. As R 0, c Ao - cAo, and 17q, - that for a PFR without recycle. An analytical proof of the second limiting case does not follow directly from these two equations. As R °°, c Ao -> cA1 (from equation 15.3-3), and V/q, -> -(w)(0) (from equation 15.3-4), which is an indeterminant form. The latter can be evaluated with the aid of L Hdpital s Rule, but the proof is left to problem 15-18. 

The same result is obtained from (105) evaluating the indeterminate form in the usual manner. 

For k = 0, —00, +00, indeterminate forms appear, most easily resolved by working with the logarithm ... 

This relation is practically useful only to eliminate further occurrences of cjj. Still, the second term of the spatial derivative, (l/r)(0c/9r), implies an indeterminate form of the type 0/0. Making use of l Hospital s rule results in the following representation ... 

Head.—An indeterminate form of inflorescence, as seen in the Daisy family, in which the flowers are in a dense duster on the receptacle. 

Provided the magnitude of the reaction terms involving can be estimated, Eqs. (13-49) and (13-50) can be solved step by step to obtain the conversion. The first step is to compute values of T and x across the diameter, at z = 1 Az, 01 L = 1, from known values at L = 0. Then continue to the next longitudinal increment, L = 2, etc. The indeterminate form of the equations at n = 0 can be avoided by using the special expressions... 

The Clapeyron equation cannot apply because both AW and AS are zero through a second-order transition, resulting in an indeterminate form 0/0. 

The value of a function is called an indeterminate form at some point a if its linut as X a apparently approaches one of the forms 0/0, oo/oo, orO-oo. Two examples are the combinations sinx/x and xhix as x 0. (As we used to say in high school, such sick functions had to be sent to T Hospital to be cured.) To be specific, let us consider a case for which... 

The indeterminate form of equation (8.27) should be noted for n = 1, but then kg is redundant for Newtonian fluids. 

To prove that (40) is an indeterminate form as A i —> 1, it will suffice to consider the expression... 

Since the product of the roots in (43) is proportional to the constant term, then it follows that ct)ia)2 - contains — 1 as a factor. Hence the numerator of (41) contains — 1 as a factor and the expression (41) is an indeterminate form. 

The indeterminate form (41) is readily evaluated with the aid of L Hos-pital s Theorem [12]. Thus, for k = I equation (40) takes the form... 

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