The difference between a runabout and a shuttlecraft is a matter of size and capability. Most of the time, their duties were interchangeable so either ship could be used. With the ability to customize ships in the Federation, there are numerous designs for shuttlecraft which includes vehicles such as the Delta Flyer (VOY) or the Captain's Yacht, Cousteau (TNG).


Shuttlecraft were small ships assigned to starships, used primarily in cargo transportation, short trips for species which experience transporter difficulties or anxiety, or movement through electromagnetic phenomenon which might disrupt or challenge transporter technologies. 
Shuttlecraft had to be fit during the TOS period with technologies to allow them to be used with warp drives. By the TNG series, shuttlecraft came standard with relatively short-hop warp capacities (some light years allowing them to be used as exploration vehicles). There were many different classes of shuttlecraft depending on the duties being carried out.
Runabouts were slightly larger than shuttlecraft but smaller than starships. Runabouts were warp-capable, had personal transporters and usually had a crew of four personnel. Properly fitted, they could function as transports, scientific vehicles or medical evacuation vehicles. Runabouts were used primarily on space stations such as Deep Space Nine (DS9) as a means of travel between the Federation, Cardassian and Bajoran sectors of space.




Runabout specifications included deflectors and phaser arrays. But they were not meant to sustain much damage, despite the fact they could be used in combat.



When to use a runabout or a shuttle in a mission?
Define 

$$ h_n (x) = x^{1 + {1 \over 2n - 1}}$$ for $x \in [-1,1]$. 

Why is $\lim_{n \to \infty} h_n(x) = |x|$? I know that for $C &gt; 0$ it holds that 
$$ \lim_{n \to \infty} C^{1\over n} = 1$$

but I don't know why the other limit should hold. Especially not 
$$ \lim_{n \to \infty} C^{1\over n} = -1$$
if $C &lt; 0$.
