Q.$\displaystyle\lim_{x\to 0}\left(\frac{x\cdot\log(1+3x)}{(e^{3x}-1)^2}\right)=$ (A) $\dfrac{1}{e^9}$ (B) $\dfrac{1}{e^3}$ (C) $\dfrac{1}{9}$ (D) $\dfrac{1}{3}$

This family of results governs how quickly an exponential or logarithmic expression changes near a reference point, without requiring their proofs to be repeated each time. The central results, taken as given, are that the limit as x tends to zero of (a to the x, minus 1) divided by x equals the natural logarithm of a for any base a greater than zero; that the limit as x tends to zero of (1 plus x) raised to the power (1 over x) equals the number e; and that the limit as x tends to zero of the natural logarithm of (1 plus x) divided by x equals 1. Each of these has a version scaled by a nonzero constant p inside the argument, which behaves the same way once the scaling is accounted for. In practice, a limit combining several exponential or logarithmic terms is solved by algebraically isolating each pie …