Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{1+\text{cosec}^3\text{x}}{\cot^2\text{x}}$
$\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{1+\text{cosec}^3\text{x}}{\cot^2\text{x}}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{1+\text{cosec}^3\text{x}}{\cot^2\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{(1+\text{cosec}\text{x})\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{ x}\big)}{\big(\text{cosec}^2\text{x}-1\big)}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{(\text{cosec}\text{x}+1)\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{x}\big)}{(\text{cosecx}-1)\big(\text{cosecx+1}\big)}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{ x}\big)}{(\text{cosec }\text{x}-1)}$
$=\frac{1+\text{cosec}^2\frac{3\pi}{2}-\text{cosec}\frac{3\pi}{2}}{\text{cosec}\frac{3\pi}{2}-1}$
$=\frac{1+(-1)^2-(-1)}{(-1)-1}$ $\Big[\therefore\text{cosec}\frac{3\pi}{2}=-1\Big]$
$=\frac{1+1+1}{-2}$
$=\frac{-3}{2}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{(1+\text{cosec}\text{x})\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{ x}\big)}{\big(\text{cosec}^2\text{x}-1\big)}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{(\text{cosec}\text{x}+1)\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{x}\big)}{(\text{cosecx}-1)\big(\text{cosecx+1}\big)}$
$=\lim\limits_{\text{x}\rightarrow\frac{3\pi}{2}}\frac{\big(1+\text{cosec}^2\text{x}-\text{cosec}\text{ x}\big)}{(\text{cosec }\text{x}-1)}$
$=\frac{1+\text{cosec}^2\frac{3\pi}{2}-\text{cosec}\frac{3\pi}{2}}{\text{cosec}\frac{3\pi}{2}-1}$
$=\frac{1+(-1)^2-(-1)}{(-1)-1}$ $\Big[\therefore\text{cosec}\frac{3\pi}{2}=-1\Big]$
$=\frac{1+1+1}{-2}$
$=\frac{-3}{2}$
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