L’Hospital rule

L Hospital rule states that if two functions, f(x) and g(x), hoth approach zero as x approaches zero, the ratio / (x) /g(x) can be evaluated as the ratio of the slopes of the curves... 

Show that L Hospital rule gives for the limit of the osmotic coefficient. 

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... 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Equation (8.12) becomes indeterminate at the centerline since both r and da/dr go to zero. Application of L Hospital s rule gives a special form for r = 0 ... 

Hint, the range is F < tfuu < 2F. Determining one of these limits is an easy exercise using L Hospital s rule. 

Now consider the limit as t approaches zero. L Hospital s rule shows that lim A/M -e ] = A/t... 

The Laplace transform of the impulse function is obtained easily by taking the limit of the unit rectangular function transform (2-20) with the use of L Hospital s rule ... 

In order to study the implications of Equation 2, it was evaluated at 80 points in the range xl = 0 to it. At xl 0, L Hospital s rule from calculus was needed. For larger xl, Equation 3 was evaluated for each xl using trapezoid rule numerical integration, yielding values for use in Equation 2. It was found that the rate of deposition is the highest for xl near zero, diminishing to zero at xl = ir. ... 

MEISENHEIMER ADDUCT LEWIS BASICITY DONOR NUMBER HARD BASES LEWIS BASE NUCLEOPHILICITY SOFT BASES L HOPITAL S RULE L Hospital s rule L HOPITAL S RULE... 

This is called a point-spread function, because it describes how what should be a point focus by geometrical optics is spread out by diffraction. The expression in the curly brackets is the one that is of interest. The other terms are phase and overall amplitude terms, as are usual with Fraunhofer diffraction expressions. The function Ji is a Bessel function of the first kind of order one, whose values can be looked up in mathematical tables. 2Ji(x)/x, the function in the curly brackets, is known as jinc(x). It is the axially symmetric equivalent of the more familiar sinc(x) = sin(x)/x (Hecht 2002), the diffraction pattern of a single slit, usually plotted in its squared form to represent intensity. Just as sinc(x) has a large central maximum, and then a series of zeros, so does jinc(x). Ji(x) = 0, but by L Hospital s rule the value of Ji(x)/x is then the ratio of the gradients, and jinc(0) = 1. The next zero in Ji(x) occurs when x = 3.832, and so that gives the first zero in jinc(x). This occurs at r = (3.832/n) x (q/2a)Xo in (3.2), which is the origin of the numerical factor in (3.1). 

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 ... 

L Hospital s rule is applicable to the first two forms, namely, 0/0 or The rest of the forms can be rewritten in terms of the first two. 

For limits that yield 0/0 or oo/oo, L Hospital s rule states... 

Logarithms of the Irmrts can be used to find the limits involving exponents using L Hospital s rule. 

As noted in the solution to Problem 13.2, when the solvent rate is lowered to a value at which the entering gas composition approaches equilibrium with the exit solvent, the required stages approach infinity. Recognize that this is not based on Equation (13.8), where it might appear that when A = 1, N in fact, when A = 1, = N/(N + 1) by L Hospital s rule. Rather, it is based on a pinch ... 

A complication arises at r = 0 due to the 1 jr term in Equation 8.20. However, since dajdr is also zero at the origin, L Hospital s rule can be applied to give... 

Consider case ( ). Here we simply have two rows of the matrix of coefficients the same and hence det A (vv) = 0 (or det Y(w) = 0). Similarly, we have the right hand side of two of the equations in (9) or (10) the same so that the numerator determinant which is formed when a column of Ar(w)(or F(w)) is replaced by the right hand column will also give two identical rows. Hence, the numerator determinant is 0. In these cases L Hospital s rule must be used to evaluate these expressions. In the present review we investigate the case r = 2. 

The first part of the programme consists of the calculation of the % matrix elements which form the coefficients of the system of equations. % The second part of the programme, as this has been explained in section % 2.1.2, consists of the iterative application of the L Hospital s rule % for the calculation of the solution of these equations that make up the % coefficients of the family of new tenth algebraic order exponentially % fitted methods for the numerical integration of the Schrodinger type % equations. 

From the theory of exponentially-fitted methods it has been shown % that to avoid divisions by zero valued determinants we must apply % l Hospital s rule. Hence we find the appropriate derivatives of the % determinants of the matrices. The theory shows that for %... 

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