Proving that a language is not Recursive
Proving that the language is in RE is quite easy: simulate the machine on every input $w\in \Sigma^*$, in minlex order, for $|w|$ steps. If an input is accepted - accept.

Proving that the language is undecidable can be done by reduction from
$$A_{TM}=\{&lt;M,w&gt;: M\text{ accepts }w\}.$$
A possible reduction:

Given input $&lt;M,w&gt;$, construct a new TM $M'$ that works as follows: given input $x$, simulate $M$ on $w$. If $M$ accepts, accept. If $M$ rejects, reject, and of course if $M$ does not halt then so will $M'$.

Now, if $&lt;M,w&gt;\in A_{TM}$, then $M$ accepts $w$ within $t$ steps, for some $t\in \mathbb{N}$. Thus, for a word $x$ with $|x|&gt;&gt;t$ (the size of $|x|$ needed depends on the manner of simulation, but can be done with $O(t\log t)$), the simulation of $M$ on $w$ will terminate within $|x|$ steps, and $M$ will accept $w$, so $M'$ will accept $x$, and $&lt;M'&gt;\in L$.

Conversely, if $&lt;M,w&gt;\notin A_{TM}$, then $M'$ will not accept any input, so $&lt;M'&gt;\notin L$, and you are done.