2. $limlimits_{h\to0}\frac{ln(4 + h)-ln(4)}{h}$ is\na 0\nb $\frac{1}{4}$\nc 1\nd e\ne nonexistent
2. $limlimits_{h\to0}\frac{ln(4 + h)-ln(4)}{h}$ 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)) is (f^{\prime}(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}). Here, if (f(x)=\ln(x)), then (\lim_{h\rightarrow0}\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})