Question

am stuck with this question:

f belongs to L1(0,infinity)

want to prove lim_t->infinity 1/t^2* interal{0 to t }x^2 *f(x)dx=0

Answer 1

Since $f \in L_1(0,\infty)$ then $\int_0^\infty |f(x)| dx < \infty$ and then

$$\begin{array}{l} \lim _ {t \rightarrow \infty} \int_ {0} ^ {t} | f (x) | d x = A < \infty \\ (\forall e > 0) (\exists d > 0) \left(\forall t > \frac {1}{d}\right) \left\{\left| \int_ {0} ^ {t} | f (x) | d x - A \right| < e \right\} \\ - e < \int_ {0} ^ {t} | f (x) | d x - A < e \\ A - e < \int_ {0} ^ {t} | f (x) | d x < A + e \\ \frac {A - e}{t ^ {2}} < \frac {1}{t ^ {2}} \int_ {0} ^ {t} | f (x) | d x < \frac {A + e}{t ^ {2}} \\ \lim _ {t \rightarrow \infty} \frac {A - e}{t ^ {2}} \leq \lim _ {t \rightarrow \infty} \frac {1}{t ^ {2}} \int_ {0} ^ {t} | f (x) | d x \leq \lim _ {t \rightarrow \infty} \frac {A + e}{t ^ {2}} \\ 0 \leq \lim _ {t \rightarrow \infty} \frac {1}{t ^ {2}} \int_ {0} ^ {t} | f (x) | d x \leq 0 \Rightarrow 0 \leq \left| \lim _ {t \rightarrow \infty} \frac {1}{t ^ {2}} \int_ {0} ^ {t} f (x) d x \right| \leq \lim _ {t \rightarrow \infty} \frac {1}{t ^ {2}} \int_ {0} ^ {t} | f (x) | d x = 0 \\ \lim _ {t \rightarrow \infty} \frac {1}{t ^ {2}} \int_ {0} ^ {t} f (x) d x = 0 \\ \end{array}$$