No, a specie cannot undergo exponential growth forever. The growth of any specie depends on the availability of food, water and space. Since Earth is finitely big, the space is limited, and so is the amount of fresh water and food. These limiting factors form the carrying capacity of a specie. It is denoted by the English alphabet 'k' in the differential equation for population growth. When the growth of a specie exceeds 'k', the growth becomes negative. Thus putting a halt to the exponential increase of the population.