Three kinds of judgments about defining domain boundedness

Three kinds of judgments about defining domain boundedness

@ (Calculus)

Given a function, there are three ways to discuss whether it is bounded on a defined field. Dare not say common, put forward to think. Theory: if f (x) is contiguous on the definition field [A, a,] or is relaxed to neurosurgery (finite first class discontinuity), then f (x) must be bounded on [a, b].

Calculation method: The existence of a \lim_{x\rightarrow a^+}f (x) exists limx→b−f (x) Existence of a \lim_{x\rightarrow b^-}f (x) exists in a continuous limx→a+f (A, B)
then f (x) bounded within the defined field [A, b].

Operation Rule determination: bounded function ±\pm bounded function = bounded function when boundary limit does not exist (finite, no infinite, infinity is a state of difficulty) bounded x bounded = bounded

This is three seemingly useless conclusions, but it can be used to understand its usefulness.

As an example:

Discussion function f (x) = (x3−1) sinx (x2+1) |x| f (x) = \frac{(x^3-1) sinx}{(x^2+1) |x|} The boundedness on its defined field.

Analysis: This kind of look is quite simple, right.

As can be seen from this function, the definition field is (−∞,0) ∪ (0,+∞) (-\infty,0) \cup (0,+\infty).

Divided into two segments, the problem is transformed into the solution of four limits.

Limx→−∞f (x) \lim_{x\rightarrow-\infty}f (x)
Limx→+∞f (x) \lim_{x\rightarrow +\infty}f (x)
Limx→0+f (x) \lim_{x\rightarrow 0^+}f (x)
Limx→0−f (x) \lim_{x\rightarrow 0^-}f (x)

If the four limits are present, then the F (x) bounds can be indicated.

Calculated separately:

Limx→−∞f (x) =limx→−∞