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In Newton's mechanics, if we apply a constant force to an object of constant mass, the acceleration will be constant. In principle we can accelerate such an object to exceed the speed of light. Not so in relativistic mechanics. The following snippet out of the eBook Relativity 4 Engineers (the link refers to a special offer to CR4 readers) shows how to compute the acceleration for a relativistic scenario: (note that the symbol v here means 'normalized velocity', i.e., v=v/c, where v is velocity in m/s and c is the speed of light in m/s).

Here all the variables are as measured in a single inertial frame of reference, e.g., the reference system of a observer that is not accelerated in any way.
The crux of this matter is that as v increases, the acceleration drops off with the factor (1-v^2). As velocity approaches the speed of light, so that (1-v^2) approaches zero, the acceleration also approaches zero, unless the force is infinitely large (which is absurd).
This then, according to relativity theory, is why nothing can exceed the speed of light as measured in any inertial frame. (My apologies for the small font of the 'snippet' - I copied it from the eBook's PDF and could not get it to come out larger. In the PDF, of course, one can set the size to anything you want.)
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