Evaluate the following limits:

\displaystyle \textbf{Question 1: }\text{Evaluate }\lim \limits_{x\to\pi}\frac{1+\cos x}{\tan^2x}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\pi}\frac{1+\cos x}{\tan^2x}
\displaystyle =\lim \limits_{x\to\pi}\frac{(1+\cos x)\cos^2x}{\sin^2x}
\displaystyle =\lim \limits_{x\to\pi}\frac{(1+\cos x)\cos^2x}{1-\cos^2x}
\displaystyle =\lim \limits_{x\to\pi}\frac{(1+\cos x)\cos^2x}{(1-\cos x)(1+\cos x)}
\displaystyle =\lim \limits_{x\to\pi}\frac{\cos^2x}{1-\cos x}
\displaystyle =\frac{\cos^2\pi}{1-\cos\pi}
\displaystyle =\frac{(-1)^2}{1-(-1)}
\displaystyle =\frac{1}{2}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Evaluate }\lim \limits_{x\to\frac{\pi}{4}}\frac{\mathrm{cosec}^2x-2}{\cot x-1}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\frac{\pi}{4}}\frac{\mathrm{cosec}^2x-2}{\cot x-1}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{1+\cot^2x-2}{\cot x-1}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{\cot^2x-1}{\cot x-1}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{(\cot x-1)(\cot x+1)}{\cot x-1}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}(\cot x+1)
\displaystyle =\cot\frac{\pi}{4}+1
\displaystyle =1+1
\displaystyle =2.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Evaluate }\lim \limits_{x\to\frac{\pi}{6}}\frac{\cot^2x-3}{\mathrm{cosec}\,x-2}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\frac{\pi}{6}}\frac{\cot^2x-3}{\mathrm{cosec}\,x-2}
\displaystyle =\lim \limits_{x\to\frac{\pi}{6}}\frac{\mathrm{cosec}^2x-1-3}{\mathrm{cosec}\,x-2}
\displaystyle =\lim \limits_{x\to\frac{\pi}{6}}\frac{\mathrm{cosec}^2x-4}{\mathrm{cosec}\,x-2}
\displaystyle =\lim \limits_{x\to\frac{\pi}{6}}\frac{(\mathrm{cosec}\,x-2)(\mathrm{cosec}\,x+2)}{\mathrm{cosec}\,x-2}
\displaystyle =\lim \limits_{x\to\frac{\pi}{6}}(\mathrm{cosec}\,x+2)
\displaystyle =2+2
\displaystyle =4.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Evaluate }\lim \limits_{x\to\frac{\pi}{4}}\frac{2-\mathrm{cosec}^2x}{1-\cot x}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\frac{\pi}{4}}\frac{2-\mathrm{cosec}^2x}{1-\cot x}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{2-(1+\cot^2x)}{1-\cot x}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{1-\cot^2x}{1-\cot x}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}\frac{(1-\cot x)(1+\cot x)}{1-\cot x}
\displaystyle =\lim \limits_{x\to\frac{\pi}{4}}(1+\cot x)
\displaystyle =1+\cot\frac{\pi}{4}
\displaystyle =1+1
\displaystyle =2.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Evaluate }\lim \limits_{x\to\pi}\frac{\sqrt{2+\cos x}-1}{(\pi-x)^2}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\pi}\frac{\sqrt{2+\cos x}-1}{(\pi-x)^2}
\displaystyle =\lim \limits_{x\to\pi}\frac{(\sqrt{2+\cos x}-1)(\sqrt{2+\cos x}+1)}{(\pi-x)^2(\sqrt{2+\cos x}+1)}
\displaystyle =\lim \limits_{x\to\pi}\frac{1+\cos x}{(\pi-x)^2(\sqrt{2+\cos x}+1)}
\displaystyle \text{Let }x=\pi-h.
\displaystyle \text{As }x\to\pi,\ h\to0.
\displaystyle =\lim \limits_{h\to0}\frac{1+\cos(\pi-h)}{h^2(\sqrt{2+\cos(\pi-h)}+1)}
\displaystyle =\lim \limits_{h\to0}\frac{1-\cos h}{h^2(\sqrt{2-\cos h}+1)}
\displaystyle =\lim \limits_{h\to0}\frac{2\sin^2\left(\frac h2\right)}{h^2(\sqrt{2-\cos h}+1)}
\displaystyle =\frac{1}{2}\lim \limits_{h\to0}\left(\frac{\sin\left(\frac h2\right)}{\frac h2}\right)^2\frac{1}{\sqrt{2-\cos h}+1}
\displaystyle =\frac{1}{2}\times1\times\frac{1}{\sqrt{2-\cos0}+1}
\displaystyle =\frac{1}{2}\times\frac{1}{2}
\displaystyle =\frac{1}{4}.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Evaluate }\lim \limits_{x\to\frac{3\pi}{2}}\frac{1+\mathrm{cosec}^3x}{\cot^2x}.
\displaystyle \text{Answer:}
\displaystyle \lim \limits_{x\to\frac{3\pi}{2}}\frac{1+\mathrm{cosec}^3x}{\cot^2x}
\displaystyle =\lim \limits_{x\to\frac{3\pi}{2}}\frac{(1+\mathrm{cosec}\,x)(1-\mathrm{cosec}\,x+\mathrm{cosec}^2x)}{\mathrm{cosec}^2x-1}
\displaystyle =\lim \limits_{x\to\frac{3\pi}{2}}\frac{(1+\mathrm{cosec}\,x)(1-\mathrm{cosec}\,x+\mathrm{cosec}^2x)}{(\mathrm{cosec}\,x-1)(\mathrm{cosec}\,x+1)}
\displaystyle =\lim \limits_{x\to\frac{3\pi}{2}}\frac{1-\mathrm{cosec}\,x+\mathrm{cosec}^2x}{\mathrm{cosec}\,x-1}
\displaystyle =\frac{1-\mathrm{cosec}\frac{3\pi}{2}+\mathrm{cosec}^2\frac{3\pi}{2}}{\mathrm{cosec}\frac{3\pi}{2}-1}
\displaystyle =\frac{1-(-1)+1}{-1-1}
\displaystyle =-\frac{3}{2}.
\displaystyle \\

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