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The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let $\\M(G)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\\Delta\\geq 3$, $\\M(G)\\le n-\\tfrac{n}{\\Delta+o(\\Delta)}$. 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