By O Bolza
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That is an adequate description from a visual point of view, but less helpful in discussing functions given by algebraic formulae. Related to this however is the idea that if we have a function f (x) then it is continuous if a small change in x results in a correspondingly small change in the value of f (x). A mathematical analysis of the intuitive idea of small changes leads to the present-day definitions used in Real Analysis. If we put these ideas together we can think of what happens to the value of f (x) as x gets closer and closer to some fixed number a, and we look for some definite number l which the values of f (x) approach.
Note that cosh is an even function, whereas sinh is odd. Just as with trigonometric functions we can define corresponding hyperbolic functions tanh x = sinh x 1 1 cosh x ; sechx = ; cosechx = . ; coth x = cosh x sinh x cosh x sinh x 1. 15. There are many identities involving hyperbolic functions, which have both similarities and differences compared with those for trigonometric functions. These can be proved using the definitions of the hyperbolic functions and the laws of indices for exponentials.
This proves that the graph does indeed oscillate between 1 and −1 infinitely many times as x approaches zero from either direction, since n can be positive or negative. So however small an interval containing zero we consider there are values of x inside that interval where f (x) = 0, where f (x) = 1 and where f (x) = −1. So f (x) does not tend to a limiting value as x tends to zero. 3 is rather erratic, because as we have shown there are infinitely many oscillations in any interval including the origin.