calculus - Definition of locally integrable function

I was given two definitions:

1. Let the function $f(x)$ be defined in a interval $[a,\infty)$ we will say that $f$ is locally integrable in $[a,\infty)$ if for all $a




2. Let $f$ be defined and locally integrable in $[a,\infty)$ we will define the improper integral $\int_{a}^{\infty}f(x)dx$ to be $\lim\limits_{R \to \infty} \int_{a}^{R}f(x)dx$.
   
   a.if the limit exist and is finite we say that $\int_{a}^{\infty}f(x)dx$ converges and $f$ is integrable in $[a,\infty)$
   
   b.if the limit does not exist we say that $\int_{a}^{\infty}f(x)dx$ diverges and $f$ is not integrable in $[a,\infty)$

Is the definition 2 part b correct?