Question

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

Explanation:

Step1: Recall the definition of the derivative

The limit $\lim_{x
ightarrow a}\frac{f(a)-f(x)}{x - a}=-\lim_{x
ightarrow a}\frac{f(x)-f(a)}{x - a}=-f^{\prime}(a)$. Here $a = 2$ and $f(x)=\ln x$.

Step2: Find the derivative of $f(x)$

The derivative of $y = \ln x$ is $y^{\prime}=\frac{1}{x}$ according to the derivative - formula for the natural - logarithm function.

Step3: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $f^{\prime}(x)$. We get $f^{\prime}(2)=\frac{1}{2}$. Then $-\lim_{x
ightarrow 2}\frac{f(x)-f(2)}{x - 2}=-f^{\prime}(2)=-\frac{1}{2}$.

Answer:

B. $-\frac{1}{2}$