Assume \( \lim _{x \rightarrow 9} f(x)=11, \lim _{x \rightarrow 9} g(x)=3 \), and \( \lim _{x \rightarrow 9} h(x)=2 \). Compute the following limit and state the limit laws used to justify the computations. \[ \lim _{x \rightarrow 9} \frac{f(x)}{g(x)-h(x)} \]

To compute the limit \( \lim _{x \rightarrow 9} \frac{f(x)}{g(x)-h(x)} \), we start by substituting the known limits into the expression. Given: \[ \lim_{x \rightarrow 9} f(x) = 11, \] \[ \lim_{x \rightarrow 9} g(x) = 3, \] \[ \lim_{x \rightarrow 9} h(x) = 2. \] We find \( \lim_{x \rightarrow 9} (g(x) - h(x)) \): \[ \lim_{x \rightarrow 9} (g(x) - h(x)) = \lim_{x \rightarrow 9} g(x) - \lim_{x \rightarrow 9} h(x) = 3 - 2 = 1. \] Now that we have both limits calculated, we can find the limit of the original expression: \[ \lim_{x \rightarrow 9} \frac{f(x)}{g(x) - h(x)} = \frac{\lim_{x \rightarrow 9} f(x)}{\lim_{x \rightarrow 9} (g(x) - h(x))} = \frac{11}{1} = 11. \] Thus, the resulting limit is: \[ \lim_{x \rightarrow 9} \frac{f(x)}{g(x)-h(x)} = 11. \] The limit laws used here are the quotient law and the difference law, which state that the limit of a quotient can be found by taking the limit of the numerator and dividing it by the limit of the denominator, provided that the limit of the denominator is not zero.