The notions of a Banach space containing asymptotically isometric copies of \(c_0\) or \(\ell^1\) first appeared in investigations of the fixed point property [Proc. Am. Math. Soc. 125, 443--446 (1997; Zbl 0861.47032)] and in the structure of classical Banach spaces [Bull. Polish Acad. Sci. Math. 48, 1--12 (2000; Zbl 0956.46006)]. In this article, the author considers asymptotically isometric analogues of results concerning Banach spaces that contain isomorphic copies of \(c_0\), \(\ell^1\), or \(\ell^\infty\). For example, \textit{W. B. Johnson} and \textit{H. Rosenthal} [Stud. Math. 43, 77--92 (1972; Zbl 0231.46035)] showed that if \(X\) is a separable Banach space whose dual contains an isomorphic copy of \(\ell^1\), then there is a quotient space of \(X\) isomorphic to \(c_0\). The author of this article proves that if \(X\) is a separable Banach space, then \(X^*\) contains an asymptotically isometric copy of \(\ell^1\) if and only if there is a quotient space of \(X\) asymptotically isometric to \(c_0\). The author also shows that, if \(X\) and \(Y\) are infinite-dimensional Banach spaces and \(Y\) contains an asymptotically isometric copy of \(c_0\), then the space \(L(X,Y)\) of bounded linear operators from \(X\) to \(Y\) contains an isometric copy of \(\ell^\infty\) (an analogue of a result of \textit{J. C. Ferrando} [Math. Scand. 77, 148--152 (1995; Zbl 0853.46031)]) and the space \(K_{w^*}(X^*,Y)\) of compact, weak*-weak continuous linear operators from \(X^*\) to \(Y\) contains a complemented, asymptotically isometric copy of \(c_0\) (an analogue of a result of \textit{R. A. Ryan} [Proc. R. Irish. Acad., Sect. A 91, 239--241 (1991; Zbl 0725.46020)]).