The ant will reach the end -- the part of the whole path he covers in a second is 0.01 / (1 + t), where t is time in seconds (the covered part doesn't decrease since the line extends uniformly). So the total part he has covered up to the time t is 0.01 * ln(1 + t), meaning he crosses whole path in exp(100) - 1 seconds (approx. 10^36 years).