Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not:
$1.$ If $K(x,t)$ is bounded but not converge with respect to positive (or negative) $x$ , then $\lim\limits_{x\to+\infty}\sum\limits_{t=a}^bf(t)K(x,t)$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\sum\limits_{t=a}^bf(t)K(x,t)\biggr)$ and $\lim\limits_{x\to+\infty}\int_a^bf(t)K(x,t)~dt$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\int_a^bf(t)K(x,t)~dt\biggr)$ should be indeterminate.
$2.$ If $\lim\limits_{x\to+\infty}K(x,t)=0$ $\biggl(\text{or}\lim\limits_{x\to-\infty}K(x,t)=0\biggr)$ , then $\lim\limits_{x\to+\infty}\sum\limits_{t=a}^bf(t)K(x,t)$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\sum\limits_{t=a}^bf(t)K(x,t)\biggr)$ and $\lim\limits_{x\to+\infty}\int_a^bf(t)K(x,t)~dt$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\int_a^bf(t)K(x,t)~dt\biggr)$ should equal to $0$ .
$3.$ If $\lim\limits_{x\to+\infty}K(x,t)=\infty$ $\biggl(\text{or}\lim\limits_{x\to-\infty}K(x,t)=\infty\biggr)$ , then $\lim\limits_{x\to+\infty}\sum\limits_{t=a}^bf(t)K(x,t)$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\sum\limits_{t=a}^bf(t)K(x,t)\biggr)$ and $\lim\limits_{x\to+\infty}\int_a^bf(t)K(x,t)~dt$ $\biggl(\text{or}\lim\limits_{x\to-\infty}\int_a^bf(t)K(x,t)~dt\biggr)$ should tend to $\infty$ .