GATE 2022 BT – Question 56
For $x_1,x_2>0$, the value of $\lim_{x_1\to x_2}(x_1-x_2)/[x_2\ln(x_1/x_2)]$ is ___________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 1
Explanation
Let $x_1=x_2(1+\varepsilon)$ with $\varepsilon\to0$.
**Numerator:** $x_1-x_2=x_2\varepsilon$.
**Denominator:**
$$x_2\ln\frac{x_1}{x_2}=x_2\ln(1+\varepsilon)=x_2\left(\varepsilon-\frac{\varepsilon^2}{2}+\cdots\right).$$
**Ratio:**
$$\frac{x_2\varepsilon}{x_2(\varepsilon-\varepsilon^2/2+\cdots)}=\frac{1}{1-\varepsilon/2+\cdots}\;\longrightarrow\;\mathbf{1}\quad(\varepsilon\to0).$$
(Using L'Hôpital's rule on $\dfrac{x_1-x_2}{\ln x_1-\ln x_2}$ with respect to $x_1$ gives $x_1\to x_2$, so the limit of the full expression is $x_2/x_2=1$.)