An Elementary Treatise On the Differential Calculus: Containing the Theory ...

Benjamin Williamson

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, \ { = oo .
Accordingly, in all cases the value of ' , ) I * determines • _^ L. -f/^i* mvka 00 that of • ) [ for either of the indeterminate forms - or oo ^ (a) o 93. Indeterminate ExpressionB of the Form {f{x)] ^^'K Let u = {/{x) ] *!^\ then log w = ^ {x) hgf{x). This latter product is indeterminate, whenever one of its factors becomes ^ero and the other infinite for the same value of x.
* On referring to Art. 69, the student will observe ihoi—-!- is of the form yw *^ ^ whenever , [ • = ...S-i «> that the process given above would not seem to assist us towards determining the true value of the fraction in this case ; how- •ever, we generally find a common factor, or else some simple transformation, by which we are enabled to exhibit our expression after differentiation in the form - .
For example, . . is of the form z~ when a; = — ; here /' {x) log (r-^j « sec 'a?, ^' {x) = — , and the fraction \) I is still of the form ^ , but it *"7 o •can be transformed into ^ which is of the form - ; the true value of the cos H o latter fraction can be easily shown to be - «> when « = - .

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