Question

lim(x→0) (5^x - 1)/x = a 0 b ln 5 c 5 ln 5 d 1

Explanation:

Step1: Recall the limit - definition of the derivative

The derivative of a function \(y = f(x)\) is defined as \(f^\prime(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\). For the function \(y = a^{x}\), \(f(x)=a^{x}\), and \(f(x + h)=a^{x + h}=a^{x}\cdot a^{h}\). Then \(f^\prime(x)=\lim_{h
ightarrow0}\frac{a^{x + h}-a^{x}}{h}=a^{x}\lim_{h
ightarrow0}\frac{a^{h}-1}{h}\). When \(x = 0\), the derivative of \(y=a^{x}\) at \(x = 0\) is \(\lim_{h
ightarrow0}\frac{a^{h}-1}{h}\).

Step2: Identify the function and its derivative

The derivative of the function \(y = a^{x}\) with respect to \(x\) is \(y^\prime=a^{x}\ln a\). Evaluating the derivative at \(x = 0\), we have \(y^\prime|_{x = 0}=a^{0}\ln a=\ln a\). In the given limit \(\lim_{x
ightarrow0}\frac{5^{x}-1}{x}\), comparing with the formula \(\lim_{h
ightarrow0}\frac{a^{h}-1}{h}\), where \(a = 5\), the value of the limit is \(\ln5\).

Answer:

B. \(\ln5\)