Question

For $x_1>0$ and $x_2>0$, the value of $\displaystyle \lim_{x_1 \to x_2} \frac{x_1 - x_2}{x_2 \ln\left(\frac{x_1}{x_2}\right)}$ is _________.

Hint:

A very useful standard limit is $\displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$, which simplifies many logarithmic limit problems.

Answer 1

Correct Answer: 1

We need to evaluate the limit: \[ L = \lim_{x_1 \to x_2} \frac{x_1 - x_2}{x_2 \ln\left(\frac{x_1}{x_2}\right)}. \] Step 1: Substitute a substitution to simplify the expression.
Let: \[ x_1 = x_2 (1 + h), \] where \( h \to 0 \) as \( x_1 \to x_2 \).
Then, \[ x_1 - x_2 = x_2 h, \] and \[ \ln\left(\frac{x_1}{x_2}\right) = \ln(1 + h). \] Step 2: Substitute into the limit expression:
\[ L = \lim_{h \to 0} \frac{x_2 h}{x_2 \ln(1+h)} = \lim_{h \to 0} \frac{h}{\ln(1+h)}. \] Step 3: Use the known standard limit:
\[ \lim_{h \to 0} \frac{\ln(1+h)}{h} = 1. \] Therefore, \[ \lim_{h \to 0} \frac{h}{\ln(1+h)} = 1. \] Step 4: Thus, the required limit is: \[ \boxed{1}. \]