JEE Mathematics 2025 Question with Solution

Step-by-step solution

Common mistakes

• Approximating both exponential terms separately by $(3x)^{x/2}$ and concluding their ratio is $1$ is incomplete. The bases differ by lower-order terms, and when raised to power $x/2$ they produce a nontrivial exponential factor. First combine them as $\left(\frac{3x-1}{3x+2}\right)^{x/2}$ and then apply the standard exponential limit.

• Cancelling only the leading polynomial terms and stopping at $\frac{2}{3}$ misses the contribution from the power term. The polynomial part tends to $\frac{2}{3}$, but the exponential part tends to $\frac{1}{\sqrt{e}}$, so both factors must be multiplied.

• Using $\left(1+u_x\right)^x \to e$ without checking the limit of $x u_x$ leads to the wrong exponent. Here $u_x = -\frac{3}{3x+2}$, so $x u_x \to -1$, and because the power is $x/2$, the limit becomes $e^{-1/2}$, not $e^{-1}$ or $1$.