Limits to selection and Mendelian sampling (MS) terms can be calculated using haplotypes by summing the individual additive effects on each chromosome. Haplotypes were imputed for 43 382 single-nucleotide polymorphisms (SNP) in 1455 Brown Swiss, 40 351 Holstein and 4064 Jersey bulls and cows using the Fortran program findhap.f90, which combines population and pedigree haplotyping methods. Lower and upper bounds of MS variance were calculated for daughter pregnancy rate (a measure of fertility), milk yield, lifetime net merit (a measure of profitability) and protein yield assuming either no or complete linkage among SNP on the same chromosome. Calculated selection limits were greater than the largest direct genomic values observed in all breeds studied. The best chromosomal genotypes generally consisted of two copies of the same haplotype even after adjustment for inbreeding. Selection of animals rather than chromosomes may result in slower progress, but limits may be the same because most chromosomes will become homozygous with either strategy. Selection on functions of MS could be used to change variances in later generations.