Question

prove that if a∫b f(x)dx exists as a proper integral ,then limx∫b f(t)dt=0.
x→b-

Answer 1

According to necessary condition of existence of proper integral

$$\left| f (x) \right| \leq C \forall x \in [ a, b ]$$

So,

$$\lim _ {x \to b ^ {-}} \int_ {x} ^ {b} f (t) d t \leq \lim _ {x \to b ^ {-}} \int_ {x} ^ {b} | f (t) | d t \leq \lim _ {x \to b ^ {-}} \int_ {x} ^ {b} C d t = \lim _ {x \to b ^ {-}} C (b - x) = 0$$

Having used the property of direction of integration

$$\lim _ {x \to b ^ {-}} \int_ {x} ^ {b} f (t) d t = - \lim _ {x \to b ^ {-}} \int_ {b} ^ {x} f (t) d t \geq - \lim _ {x \to b ^ {-}} \int_ {b} ^ {x} | f (t) | d t \geq - \lim _ {x \to b ^ {-}} \int_ {b} ^ {x} C d t = - \lim _ {x \to b ^ {-}} C (x - b) = 0$$

Finally we have

$$\left\{ \begin{array}{l} \lim _ {x \to b ^ {-}} \int_ {x} ^ {b} f (t) d t \leq 0 \\ \lim _ {x \to b ^ {-}} \int_ {x} ^ {b} f (t) d t \geq 0 \end{array} \right. \Rightarrow \lim _ {x \to b ^ {-}} \int_ {x} ^ {b} f (t) d t = 0$$