Let \(\beta\in(1,2)\) and \(x\in [0,1/(\beta-1)]\). We call a sequence \((\varepsilon_{i})_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}\) a \(\beta\)-expansion for \(x\) if \(x=\sum_{i=1}^{\infty}\varepsilon_{i}\beta^{-i}\). We call a finite sequence \((\varepsilon_{i})_{i=1}^{n}\in\{0,1\}^{n}\) an \(n\)-prefix for \(x\) if it can be extended to form a \(\beta\)-expansion of \(x\). In this paper we study how good an approximation is provided by the set of \(n\)-prefixes. Given \(\Psi:\mathbb{N}\to\mathbb{R}_{\geq 0}\), we introduce the following subset of \(\mathbb{R}\): \[ W_{\beta}(\Psi):=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}\bigcup_{(\varepsilon_{i})_{i=1}^n\in\{0,1\}^{n}}\biggl[\sum_{i=1}^n\frac{\varepsilon_i}{\beta^i},\sum_{i=1}^{n}\frac{\varepsilon_i}{\beta^i}+\Psi(n)\biggr] \] In other words, \(W_{\beta}(\Psi)\) is the set of \(x\in\mathbb{R}\) for which there exist infinitely many solutions to the inequalities \[ 0\leq x-\sum_{i=1}^{n}\frac{\varepsilon_i}{\beta^i}\leq \Psi(n). \] When \(\sum_{n=1}^{\infty}2^n\Psi(n)<\infty\), the Borel-Cantelli lemma tells us that the Lebesgue measure of \(W_{\beta}(\Psi)\) is zero. When \(\sum_{n=1}^{\infty}2^{n}\Psi(n)=\infty,\) determining the Lebesgue measure of \(W_{\beta}(\Psi)\) is less straightforward. Our main result is that whenever \(\beta\) is a Garsia number and \(\sum_{n=1}^{\infty}2^{n}\Psi(n)=\infty\) then \(W_{\beta}(\Psi)\) is a set of full measure within \([0,1/(\beta-1)]\). Our approach makes no assumptions on the monotonicity of \(\Psi,\) unlike in classical Diophantine approximation where it is often necessary to assume \(\Psi\) is decreasing.