Question

1. $lim_{h \to 0} \frac{ln(4 + h) - ln(4)}{h}$ 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 \(y = f(x)\) at \(x=a\) is \(f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\). Here, if \(f(x)=\ln(x)\) and \(a = 4\), then \(\lim_{h
ightarrow0}\frac{\ln(4 + h)-\ln(4)}{h}=f^{\prime}(4)\).

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

The derivative of \(y=\ln(x)\) using the formula \((\ln x)^{\prime}=\frac{1}{x}\). So when \(x = 4\), \(f^{\prime}(x)=\frac{1}{x}\) gives \(f^{\prime}(4)=\frac{1}{4}\).

Answer:

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