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Name: Alan
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Question:
The Richter Scale yields very large numerical values from powers of base 10. If know that the value is equal to an exponent of base ten, how do I calculate the exponent without factoring the numerical value by 10?

Example: 10 to the 8.6 power equals 398107170.6. 10 to the X power equals 398107170.6. How do I find X without factoring?



Replies:
To be sure that the expression is correct, I will use the symbol "^" to mean "raise to the power". Then, the limit you seek is: lim x----> infinity [(x^2 -1)^1/2] / x This meets on of the criteria for applying L'Hospital's rule, namely that the ratio is indeterminate of the form: [infinity] / [infinity]. Now here is where the rule is most often confused. The ratio is of the form: N(x) / D(x) where, Numerator N(x) = (x^2 -1)^1/2 and the Denominator D(x) = x. You must now determine the derivatives of N(x) and D(x) SEPARATELY and INDIVIDUALLY: So, dN(x)/dx = [(x^2 - 1)^-1/2] * (x) and dD(x)/dx = 1. Then: the lim x ----> infinity [ x / (x^2 - 1)^1/2 ] / [1] which "by inspection" without a detailed proof is [ 1 ] / [ 1 ] = 1 as you recognized. The SEPARATELY and INDIVIDUALLY comes from the definition of the derivative which has in the denominator [(x + a) - a] which is identically the same for N(x) and D(x) and so cancels out in the quotient.

An added word of caution, L'Hospital's rule must be very carefully applied to damped trigonometric functions because the damping term may make the limit appear to be finite whereas it actually continues to oscillate, but with a large damping factor. See also the NEWTON archive:

http://www.newton.dep.anl.gov/webpages/askasci/math99/math99169.htm

Vince Calder


Hello,

L'Hospital rule is used for cases that reduce to an undeterminable condition of zero/zero in the limit. In your example, you do not have such a condition. Dividing through by x and letting x approach infinity, you obtain the answer. Alternatively, replacing x by 1/u, and letting u approach zero result in the same finite answer without the need to apply that rule.

Ali Khounsary, Ph.D.
Advanced Photon Source
Argonne National Laboratory
Argonne, IL


Mike,

L'Hospital's rule does apply, but the result just does not produce a convenient result. One useful method is to find the limit of the square of the function. That would be (x squared - 1)/(x squared). Now use L'Hospital's rule. Since the limit of the function-squared equals 1, then the limit of the function itself is also 1.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College



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