Q.$\displaystyle\lim_{x\to 0}\left[\frac{\log(5+x)-\log(5-x)}{\sin x}\right]=$ (A) $\dfrac{3}{2}$ (B) $-\dfrac{5}{2}$ (C) $-\dfrac{1}{2}$ (D) $\dfrac{2}{5}$

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 …