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Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits2 Marks
Question
Evaluate the following limit: $\lim\limits_{{\text{x}}\rightarrow\frac{\pi}{2}}\frac{\text{a}^{\cot\text{x}}-\text{a}^{\cos\text{x}}}{\cot\text{x}-\cos\text{x}}$
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Answer

$\lim\limits_{{\text{x}}\rightarrow\frac{\pi}{2}}\frac{\text{a}^{\cot\text{x}}-\text{a}^{\cos\text{x}}}{\cot\text{x}-\cos\text{x}}$ $=\lim\limits_{{\text{x}}\rightarrow\frac{\pi}{2}}\text{a}^{\cos\text{x}}\Big[\frac{\text{a}^{\cot\text{x}-{\cos\text{x}}}-1}{\cot\text{x}-\cos\text{x}}\Big]$ $=1\times\text{log a}$ $=\text{log a}$

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