Limits, Continuity & Differentiability
Limits Involving Special Functions
Grade 12
<p>Let $[\cdot]$ represent the greatest integer function less than or equal to $x$. The value of $\lim_{x \to 0} \frac{\ln \sin x}{[\ln \tan x]} - \frac{\ln \tan x}{[\ln \tan x]}$ is</p>
Step-by-Step Solution
Key Concept: Analyze the behavior of logarithmic and greatest integer functions as $x \to 0^+$. Use standard limit relationships between $\sin x$, $\tan x$, and their logarithms.
<p>As $x \to 0^+$: $\tan x \to 0^+$, so $\ln \tan x \to -\infty$, and $[\ln \tan x] \to -\infty$.</p><p>Also, $\sin x \to 0^+$, so $\ln \sin x \to -\infty$.</p><p>The ratio $\frac{\ln \sin x}{\ln \tan x} = \frac{\ln \sin x}{\ln(\sin x/\cos x)} \to 1$ as $x \to 0^+$.</p><p>Therefore, $\lim_{x \to 0} \left(1 - 1\right) = 0$.</p>
Correct Answer: A