11.04 - Paradox 0 = ln 1 = ln(1(1 = ln(1 ln(1 = undefined...

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Paradox: 0 = ln 1 = ln ((–1)/(–1)) = ln (–1) – ln (–1) = undefined! What’s going on? … ..?.. The rule ln a / b = ln a – ln b only says that if ln a and ln b are defined (that is, if a and b are positive), then ln a / b = ln a – ln b . It says nothing if ln a is undefined and/or ln b is undefined. Prologue to section 3.7: Limits of ratios How can we determine the limiting behavior of f ( x )/ g ( x ) in terms of the limiting behavior of f ( x ) and the limiting behavior of g ( x )? For simplicity, let’s assume f ( x ) and g ( x ) are positive for all x . Also, we’ll look at the limiting behavior as x →∞ (though the story for limits of the form x a , with a finite, is much the same). We’ll distinguish between three kinds of functions: those that (like 1/ x or 1/ x 2 ) go to 0+ as x →∞ ; those that go to some positive number a as x →∞ , and those that (like sqrt( x ) or ln x ) go to + as x →∞ . (There are function that exhibit none of these behaviors, like sin x , but we’ll ignore them
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for now.) Here’s a chart showing what lim x →∞ f ( x )/ g ( x ) could be, given lim x →∞ f ( x ) and lim x →∞ g ( x ); the rows specify values of the former limit, and the columns specify values of the latter limit. 0+ b >0 + 0+ ind. 0 0 a >0 + a / b 0 + + + ind. The cases of “0+/0+” and “+ /+ ” are indeterminate , in the sense that “more information is required”; we need a new technique to handle them.
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