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The first three functions have points for which the limit does not exist, while the function = is not defined at =, but its limit does exist. respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. [7] If the one-sided limits exist at p, but are unequal, then there is no limit at ...
The adjective indeterminate does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.
The function in example 2, a jump discontinuity. Consider the function = {< = > Then, the point = is a jump discontinuity.. In this case, a single limit does not exist because the one-sided limits, and + exist and are finite, but are not equal: since, +, the limit does not exist.
This means that if |g(x)| diverges to infinity as x approaches c and both f and g satisfy the hypotheses of L'Hôpital's rule, then no additional assumption is needed about the limit of f(x): It could even be the case that the limit of f(x) does not exist. In this case, L'Hopital's theorem is actually a consequence of Cesàro–Stolz.
In some cases in which the limit does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as approaches is sometimes called a "two-sided limit". [citation needed] It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided ...
This fails if J is not connected. For example, if J is a discrete category, ... (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.)
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]