Transcription of Indeterminate Forms - Florida State University
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Indeterminate Forms - Florida State University
Indeterminate Forms00, ,0 ,00, 0,1 , These are the so called Indeterminate Forms . One can apply L Hopital s rule directly to the forms00and . It is simple to translate 0 into01/ or into 1/0, for example one can write limx xe xaslimx x/exor as limx e x/(1/x). To see that the exponent Forms are Indeterminate note thatln 00= 0 ln 0 = 0( ) = 0 ,ln 0= 0 ln = 0 ,ln 1 = ln 1 = 0 = 0 These formula s also suggest ways to compute these limits using L Hopital s rule. Basically we use twothings, thatexand lnxare inverse functions of each other, and that they are continuous functions. If g(x)is a continuous function theng(limx af(x)) = limx ag(f(x)).For example let s figure out limx (1 +1x)x= is of the Indeterminate form 1 . We write exp(x)forexso to reduce the amount (1 +1x)x= exp(ln( limx (1 +1x)x)) = exp( limx ln((1 +1x)x))= exp( limx xln(1 +1x)) = exp( limx ln(1 +1x)1/x)We can now apply L Hopital s since the limit is of the exp( limx (1/(1 +1x))( 1/x2) 1/x2) = exp( limx 1/(1 +1x)) = exp(1) = the (1 +1x) (1 +kx) 0(1 +x)1 0+ 0+x(x2) 0+x1 might be tempted to handle in a similar manner sincee =e e =.
Indeterminate Forms 0 0, ∞ ∞, 0 ·∞, 00, ∞0, 1∞, ∞−∞ These are the so called indeterminate forms. One can apply L’Hopital’s rule directly to the forms 0 0
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