Mathematical Methods for Physics and Engineering - Matematica.NET

4.7 EVALUATION OF LIMITSThese can all be derived by straightforward application of Taylor’s theorem tothe expansion of a function about x =0.4.7 Evaluation of limitsThe idea of the limit of a function f(x) asx approaches a value a is fairly intuitive,though a strict definition exists and is stated below. In many cases the limit ofthe function as x approaches a will be simply the value f(a), but sometimes thisis not so. Firstly, the function may be undefined at x = a, as, for example, whenf(x) = sin xx ,which takes the value 0/0 atx = 0. However, the limit as x approaches zerodoes exist and can be evaluated as unity using l’Hôpital’s rule below. Anotherpossibility is that even if f(x) is defined at x = a its value may not be equal to thelimiting value lim x→a f(x). This can occur for a discontinuous function at a pointof discontinuity. The strict definition of a limit is that if lim x→a f(x) =l thenfor any number ɛ however small, it must be possible to find a number η such that|f(x)−l|