91Ó°ÊÓ

Recall the following basic limit properties: 1. \(\lim _{x \rightarrow a} [c * f(x)] = c * \lim _{x \rightarrow a} f(x)\), where c is a constant. 2. \(\lim _{x \rightarrow a} [f(x) \pm g(x)] = \lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)\) 3. \(\lim _{x \rightarrow a} [f(x) * g(x)] = \lim _{x \rightarrow a} f(x) * \lim _{x \rightarrow a} g(x)\) 4. \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}\), provided \(\lim _{x \rightarrow a} g(x) \neq 0\).

We are given that \(\lim _{x \rightarrow a} f(x)=3\), and \(\lim _{x \rightarrow a} g(x)=4\). We want to find the limit of the function \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)}\). Using the basic limit properties mentioned in Step 1, we can break down the limit of the given function: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{\lim _{x \rightarrow a} (2 f(x)-g(x))}{\lim _{x \rightarrow a} (f(x) g(x))}\) We can further break down the limit: \(\lim _{x \rightarrow a} (2 f(x)-g(x)) = 2 \lim _{x \rightarrow a} f(x) - \lim _{x \rightarrow a} g(x)\) \(\lim _{x \rightarrow a} (f(x) g(x)) = \lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)\) Now, plug the given limits for f(x) and g(x) into the equation: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{2(3) - 4}{3 \cdot 4}\)

Now we simply need to simplify our expression and find the limit: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{6 - 4}{12} = \frac{2}{12}\) Simplifying our fraction, we can get: \(\lim _{x \rightarrow a} \frac{2 f(x)-g(x)}{f(x) g(x)} = \frac{1}{6}\) So, the requested limit is \(\boxed{\frac{1}{6}}\).