Abstract
Abstract “Transcendental Functions” examines logarithmic, exponential, inverse trig, and hyperbolic functions, among others. The traditional definition of the natural logarithm as a definite integral is used. Several sections of fairly standard material follow, with the exception that limits of these functions are computed using hyperreal numbers. The diagram of levels of hyperreal numbers is updated to include these new functions, culminating in its use for easy comparisons of rates of growth with application to big-oh notation. This system also reduces (but does not eliminate) the need for l’Hospital’s rule, which is presented along with a discussion of various indeterminate forms. The chapter ends with an optional discussion of nonelementary functions developed in the same manner as the natural logarithm. The sine integral function is used as an example, and others are explored in the exercises, with the purpose of alerting readers to the existence of functions beyond those studied in calculus.
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