  We prove that the number of distinct homotopy types of limits of
one-parameter semi-algebraic families of closed and bounded semi-algebraic sets
is bounded singly exponentially in the additive complexity of any
quantifier-free first order formula defining the family. As an important
consequence, we derive that the number of distinct homotopy types of
semi-algebraic subsets of $\mathbb{R}^k$ defined by a quantifier-free first
order formula $\Phi$, where the sum of the additive complexities of the
polynomials appearing in $\Phi$ is at most $a$, is bounded by
$2^{(k+a)^{O(1)}}$. This proves a conjecture made by Basu and Vorobjov [On the
number of homotopy types of fibres of a definable map, J. Lond. Math. Soc. (2)
2007, 757--776].
