if (f(x)=ln x), then (lim_{x\rightarrow2}\frac{f(2)-f(x)}{x - 2}=) a (-ln2) b (-\frac{1}{2}) c (\frac{1}{2})…

Explanation:

Step1: Find (f(2))

Given (f(x)=\ln x), then (f(2)=\ln 2).

Step2: Rewrite the limit

The limit (\lim_{x\rightarrow2}\frac{f(2)-f(x)}{x - 2}=-\lim_{x\rightarrow2}\frac{f(x)-f(2)}{x - 2}).

Step3: Recall the definition of the derivative

The definition of the derivative of a function (y = f(x)) at (x = a) is (f^{\prime}(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a}). Here (a = 2) and (f(x)=\ln x), and the derivative of (y=\ln x) is (f^{\prime}(x)=\frac{1}{x}). So (f^{\prime}(2)=\frac{1}{2}).

Step4: Calculate the original - limit

Since (\lim_{x\rightarrow2}\frac{f(2)-f(x)}{x - 2}=-\lim_{x\rightarrow2}\frac{f(x)-f(2)}{x - 2}=-f^{\prime}(2)), substituting (f^{\prime}(2)=\frac{1}{2}), we get (-\frac{1}{2}).

Answer:

B. (-\frac{1}{2})