2. $lim_{h \to 0} \frac{ln(4 + h) - ln(4)}{h}$ is\na 0\nb $\frac{1}{4}$\nc 1\nd e\ne nonexistent
2. $lim_{h \to 0} \frac{ln(4 + h) - ln(4)}{h}$ is\na 0\nb $\frac{1}{4}$\nc 1\nd e\ne nonexistent
Answer
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\rightarrow0}\frac{f(a + h)-f(a)}{h}). Here, if (f(x)=\ln(x)) and (a = 4), then (\lim_{h\rightarrow0}\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})