Q203 / 270 · NDA 2023 (Phase 2) · Mathematics
Let $f(x) \quad \frac{a^{x-1} b^{x-1}}{}$ and $g(x)=x-1$.
What is $\lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to?
- A$\frac{\ln (a b)}{4}$
- Correct answerB$\frac{\ln (a b)}{2}$
- C$\ln (a b)$
- D$2 \ln (a b)$
Solution
Since, $\lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$
$\begin{aligned}
& =\lim _{x \rightarrow 1} \frac{\frac{a^{x-1}+b^{x-1}}{2}-1}{x-1}=\lim _{x \rightarrow 1} \frac{a^{x-1}+b^{x-1}-2}{2(x-1)} \\
& =\lim _{x \rightarrow 1}\left(\frac{a^{x-1}-1}{2(x-1)}+\frac{b^{x-1}-1}{2(x-1)}\right)=\frac{1}{2} \ln a+\frac{1}{2} \ln (b) \\
& =\frac{1}{2} \ln (a b)
\end{aligned}$
Hence, the correct answer is option: 'B'.