Question

if (f(x)=ln x), then (lim_{x
ightarrow2}\frac{f(2)-f(x)}{x - 2}=) a (-ln2) b (-\frac{1}{2}) c (\frac{1}{2}) d (ln2)

Explanation:

Step1: Find \(f(2)\)

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

Step2: Rewrite the limit

The limit \(\lim_{x
ightarrow2}\frac{f(2)-f(x)}{x - 2}=-\lim_{x
ightarrow2}\frac{f(x)-f(2)}{x - 2}\).

Step3: Recall the definition of the derivative

The definition of the derivative of a function \(y = f(x)\) at \(x = a\) is \(f^{\prime}(a)=\lim_{x
ightarrow a}\frac{f(x)-f(a)}{x - a}\). 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 \(\lim_{x
ightarrow2}\frac{f(2)-f(x)}{x - 2}=-\lim_{x
ightarrow2}\frac{f(x)-f(2)}{x - 2}=-f^{\prime}(2)\), substituting \(f^{\prime}(2)=\frac{1}{2}\), we get \(-\frac{1}{2}\).

Answer:

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