Question

If \(\underset{x\rightarrow 0}{lim}\frac{45^x-9^x-5^x+1}{(k^x-1)(3^x-1)} = 2\), then the value of \(k\) is ...

• A. \(45\)
• B. \(9\)
• C. \(5\)
• D. \(3\)

Hint:

Factorise the numerator as (9^x-1)(5^x-1) and use the limit of (a^x-1)/x.

Answer 1

The Correct Option is C

Step 1: Observe the structure
The numerator has the form $(a^x-1)(b^x-1)$ with $a=9, b=5$, since $9\times5=45$.

Step 2: Small x approximation
For small $x$, $a^x - 1 \approx x\ln a$. So the quotient is about $\frac{x\ln9\cdot x\ln5}{x\ln k\cdot x\ln3} = \frac{2\ln5}{\ln k}$.

Step 3: Solve
Set $\frac{2\ln5}{\ln k} = 2$ to get $k=5$. Option (C).

Final Answer:
k = 5. \[ \boxed{\text{(C)}\ 5} \]