Question

1. $limlimits_{h\to0}\frac{ln(4 + h)-ln(4)}{h}$ is

a 0
b $\frac{1}{4}$
c 1
d e
e nonexistent

Explanation:

Step1: Recall the definition of the derivative

The definition of the derivative of a function \(y = f(x)\) is \(f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\). Here, if \(f(x)=\ln(x)\), then \(\lim_{h
ightarrow0}\frac{\ln(4 + h)-\ln(4)}{h}\) is \(f^{\prime}(4)\).

Step2: Find the derivative of \(y=\ln(x)\)

The derivative of \(y = \ln(x)\) is \(y^{\prime}=\frac{1}{x}\) (by the formula \((\ln x)^{\prime}=\frac{1}{x}\)).

Step3: Evaluate the derivative at \(x = 4\)

Substitute \(x = 4\) into \(y^{\prime}=\frac{1}{x}\). We get \(y^{\prime}\mid_{x = 4}=\frac{1}{4}\).

Answer:

B. \(\frac{1}{4}\)