The purpose of the paper is to prove the existence of \(\lim_{s\to\infty} \exp(sA+B)\) where \(A\), \(B\) are bounded linear operators in a Banach space \(X\) and \(A\) is semistable (i.e., if \(\sigma(A)\) denotes the spectrum of \(A\) and \(H^-\) denotes the open left half-plane of the complex plane \(\mathbb{C}\), then \(\sigma(A)\subset H^-\cup\{0\}\), where \(0\) is at most a simple pole of \(A\)). The proof is based on the upper semicontinuity of the spectrum and on a uniform perturbation result for resolvents. Some applications to differential equations are given.