b) Calculate the value of \[ \lim _{x \rightarrow \infty} \frac{(1+10 x)^{-\frac{1}{3}}-1}{x} \]

To evaluate the limit \[ \lim _{x \rightarrow \infty} \frac{(1+10 x)^{-\frac{1}{3}}-1}{x}, \] we start by rewriting the expression \((1 + 10x)^{-\frac{1}{3}}\). As \(x\) approaches infinity, \(10x\) becomes very large, making \(1 + 10x\) also very large. Thus, we can analyze the limit as follows: \[ (1 + 10x)^{-\frac{1}{3}} = \frac{1}{(1 + 10x)^{\frac{1}{3}}}. \] Now as \(x \to \infty\), we have \[ (1 + 10x)^{\frac{1}{3}} \sim (10x)^{\frac{1}{3}} = 10^{\frac{1}{3}} x^{\frac{1}{3}}. \] Therefore, \[ (1 + 10x)^{-\frac{1}{3}} \sim \frac{1}{10^{\frac{1}{3}} x^{\frac{1}{3}}}. \] Substituting this back into our limit gives: \[ \lim_{x \to \infty} \frac{\frac{1}{10^{\frac{1}{3}} x^{\frac{1}{3}}} - 1}{x}. \] This can be simplified to: \[ \lim_{x \to \infty} \frac{\frac{1 - 10^{\frac{1}{3}} x^{\frac{1}{3}}}{10^{\frac{1}{3}} x^{\frac{1}{3}}}}{x} = \lim_{x \to \infty} \frac{1 - 10^{\frac{1}{3}} x^{\frac{1}{3}}}{10^{\frac{1}{3}} x^{\frac{4}{3}}}. \] Now, in evaluating \(1 - 10^{\frac{1}{3}} x^{\frac{1}{3}}\), we note that as \(x \to \infty\), this expression approaches \(-10^{\frac{1}{3}} x^{\frac{1}{3}}\) which dominates the numerator. Thus, we have: \[ \lim_{x \to \infty} \frac{-10^{\frac{1}{3}} x^{\frac{1}{3}}}{10^{\frac{1}{3}} x^{\frac{4}{3}}} = \lim_{x \to \infty} -\frac{1}{x}, \] and as \(x \to \infty\), this limit approaches \(0\). Therefore, the limit is \[ \boxed{0}. \]