MATHS 5 Marks Questions

$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\sqrt{3+\text{x}}-\sqrt{5-\text{x}}\big)}{\big(\text{x}^2-1\big)}\times\frac{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{(3+\text{x})-(5-\text{x})}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{-2+2\text{x}}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{2}{(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5+\text{x}}\big)}$

$=\frac{2}{(1+1)\big(\sqrt{3+1}+\sqrt{5-1}\big)}=\frac{2}{(2)(2+2)}$

$=\frac14$ 

As $\text{x}\rightarrow\frac{\pi}{4},\frac\pi4-\text{x}\rightarrow0,$ let $\frac{\pi}{4}-\text{x}=\text{y}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{2}-\cos\big(\frac\pi4-\text{y}\big)-\sin\big(\frac\pi4-\text{y}\big)}{\text{y}^2}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{2}-\big[\cos\frac\pi4\cos\text{y}+\sin\frac{\pi}{4}\sin\text{y}+\sin\frac\pi4\cos\text{y}-\cos\frac\pi4\sin\text{y}\big]}{\text{y}^2}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{2}-\Big(\frac{\cos\text{y}}{\sqrt{2}}+\frac{\sin\text{y}}{\sqrt{2}}+\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}\Big)}{\text{y}^2}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\frac{2\cos\text{y}}{\sqrt{2}}}{\text{y}^2}=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{2}-\sqrt{2}\cos\text{y}}{\text{y}^2}$

$=\sqrt{2}\lim\limits_{\text{y}\rightarrow{0}}\frac{(1-\cos\text{y})}{\text{y}^2}$

$=\sqrt{2}\lim\limits_{\text{y}\rightarrow{0}}\frac{2\sin^2\frac{\text{y}}{2}}{\frac{\text{y}^2}{4}}\times\frac14$

$=\sqrt{2}\times2\times\frac14\bigg(\lim\limits_{\text{y}\rightarrow0}\frac{\sin\frac{\text{y}}{2}}{\frac{\text{y}}{2}}\bigg)^2$

$=\sqrt{2}\times2\times\frac14\times1$

$=\frac{1}{\sqrt{2}}$ 

$=\lim\limits_{\text{x} \rightarrow0}\frac{3\sin^2\text{x}-2\sin\text{x}^2}{3\text{x}^2}-\lim\limits_{\text{x} \rightarrow0}\frac{2\sin\text{x}^2}{3\text{x}^2}$

$=\lim\limits_{\text{x} \rightarrow0}\big(\frac{\sin\text{x}}{\text{x}}\big)^2-\frac{2}{3}\lim\limits_{\text{x} \rightarrow0}\frac{\sin\text{x}^2}{\text{x}^2}$

$=1-\frac{2}{3}\times1$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$

$=1-\frac23$

$=\frac{3-2}{3}$

$=\frac{1}{3}$ 

$=\lim\limits_{\text{x}\rightarrow{4}}\frac{\text{x}^{3}-4^3}{\text{x}^2-4^2}$

$=\lim\limits_{\text{x}\rightarrow{4}}\frac{\frac{\text{x}^{3}-4^3}{\text{x}-4}}{\frac{\text{x}^2-4^2}{\text{x}-4}}$ [Dividing numerator and denominator by x - 4]

$=\frac{\lim\limits_{\text{x}\rightarrow4}\frac{\text{x}^3-4^3}{\text{x}-4}}{\lim\limits_{\text{x}\rightarrow4}\frac{\text{x}^2-4}{\text{x}-4}}$

Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$ in numerator and $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{m}}-\text{a}^{\text{m}}}{\text{x}-\text{a}}=\text{ma}^{\text{m}-1}$ in denominator

$\Rightarrow\text{n}=3,\text{m}=2$

$\Rightarrow\frac{\lim\limits_{\text{x}\rightarrow4}\frac{\text{x}^3-4^3}{\text{x}-4}}{\lim\limits_{\text{x}\rightarrow4}\frac{\text{x}^2-4}{\text{x}-4}}=\frac{3(4)^{3-1}}{2(4)^{2-1}}=\frac{3(4)^2}{2(4)}$

$=6$ 

$\lim\limits_{\text{x}\rightarrow4}\frac{\big(2-\sqrt{\text{x}}\big)\big(2+\sqrt{\text{x}}\big)}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$

$\lim\limits_{\text{x}\rightarrow4}\frac{(2)^2-\big(\sqrt{\text{x}}\big)^2}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$

$\lim\limits_{\text{x}\rightarrow4}\frac{(4-\text{x})}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$

$=\frac{1}{2+\sqrt{4}}$

$=\frac{1}{2+2}$

$=\frac14$ 

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\sqrt{\text{x}}-\sin\sqrt{\text{a}}}{\big(\sqrt{\text{x}}-\sqrt{\text{a}}\big)\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{2\sin\Big(\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}\Big)\cos\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)}{\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)\Big({\sqrt{\text{x}}-\sqrt{\text{a}}}\Big)}$

$=2\begin{pmatrix}\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\frac{\sqrt{\text{x}}-\sqrt{\text{a}}}{2}}{\Big(\frac{{\sqrt{\text{x}}+\sqrt{\text{a}}}}{2}\Big)}\end{pmatrix}\times\frac12\frac{\lim\limits_{\text{x}\rightarrow{\text{a}}}\cos\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{2}\Big)}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$

$=2\times1\times\frac{1}{2}\times\cos\sqrt{\text{a}}\times\frac{1}{2\sqrt{\text{a}}}$

$=\frac{\cos\sqrt{\text{a}}}{2\sqrt{\text{a}}}$

$=\lim\limits_{{\text{x}-{\frac\pi4\rightarrow0}}}\frac{\big(\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}\big)}{\big(\text{x}-\frac{\pi}{4}\big)}\times\frac{\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}{\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}$

$=\lim\limits_{{\text{x}-{\frac\pi4\rightarrow0}}}\frac{\big(\sqrt{\cos\text{x}}-\sqrt{\sin\text{x}}\big)}{\big(\text{x}-\frac{\pi}{4}\big)\big(\sqrt{\cos\text{x}}+\sqrt{\sin\text{x}}\big)}$

As $\text{x}\rightarrow\frac{\pi}{4}\Rightarrow\text{x}-\frac\pi4\rightarrow0\Rightarrow$let $\text{x}-\frac\pi4=\text{y}$

$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(\cos\big(\frac\pi4+\text{y}\big)-\sin\big(\frac\pi4+\text{y}\big)\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$

$=\lim\limits_{\text{y}\rightarrow0}\frac{\big(\cos\frac\pi4\cos\text{y}-\sin\frac{\pi}{4}\sin\text{y}\big)-\big(\sin\frac\pi4\cos\text{y}-\cos\frac{\pi}{4}\sin\text{y}\big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$

$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}-\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$

$=\lim\limits_{\text{y}\rightarrow0}\frac{\Big(-2\frac{\sin\text{y}}{\sqrt{2}}\Big)}{\text{y}\Big(\sqrt{\cos\big(\frac\pi4+\text{y}\big)}+\Big(\sqrt{\sin\big(\frac\pi4+\text{y}\big)}\Big)}$

$=-\sqrt{2}\Big(\lim\limits_{\text{y}\rightarrow0}\frac{\sin\text{y}}{\text{y}}\Big)\times\frac{1}{\lim\limits_{\text{y}\rightarrow0}\sqrt{\cos\big(\text{y}+\frac\pi4}\big)+\lim\limits_{\text{y}\rightarrow0}\sqrt{\sin\big(\text{y}+\frac{\pi}{4}\big)}}$

$=-\sqrt{2}\times1\times\frac{1}{\sqrt{\cos\frac{\pi}{4}}+\sqrt{\sin\frac\pi4}}$

$=-\sqrt{2}\times\frac{1}{\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}+\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}}$ $\Big[\because\cos\frac\pi4=\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\Big]$

$=\frac{-\sqrt{2}}{\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}+(1+1)^{\frac{1}{2}}}$

$=\frac{-\sqrt{2}}{\sqrt{2}\big(\frac{1}{\sqrt{2}}\big)^{\frac{1}{2}}}$

$=-\frac{1}{2^{\frac14}}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}{(\text{x}-1)(\text{x}+1)}\times\frac{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}{\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{((3+\text{x})-(5-\text{x}))}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{-2+2\text{x}}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{-2(\text{x}-1)}{(\text{x}-1)(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{-2}{(\text{x}+1)\big(\sqrt{3+\text{x}}+\sqrt{5-\text{x}}\big)}$

$=\frac{2}{(1+1)\big(\sqrt{3+1}+\sqrt{5-1}\big)}$

$=\frac{2}{(2)(2+2)}$

$=\frac14$

$=\lim\limits_{\text{x}\rightarrow2}\frac{\big(\sqrt{3-\text{x}}-1\big)}{2-\text{x}}\times\frac{\big(\sqrt{3-\text{x}}+1\big)}{\big(\sqrt{3-\text{x}}+1\big)}$

$=\lim\limits_{\text{x}\rightarrow2}\frac{(3-\text{x})-1}{(2-\text{x})\big(\sqrt{3-\text{x}}+1\big)}$

$=\lim\limits_{\text{x}\rightarrow2}\frac{(2-\text{x})}{(2-\text{x})\big(\sqrt{3-\text{x}}+1\big)}$

$=\lim\limits_{\text{x}\rightarrow2}\frac{1}{\big(\sqrt{3-\text{x}}+1\big)}$

$=\frac{1}{\sqrt{3-2}+1}=\frac{1}{1+1}$

$=\frac{1}{2}$

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{({\text{x}}-\text{a})\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}{\big(\sqrt{\text{x}}-\sqrt{\text{a}}\big)\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}$

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}-\text{a})-\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)}{(\text{x}-\text{a})}$

$=\lim\limits_{\text{x}\rightarrow{\text{a}}}\big(\sqrt{\text{x}}+\sqrt{\text{a}}\big)$

$=2\sqrt{\text{a}}$ 

Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^{\text{n}}}{\text{x}-\text{a}}=\text{na}^{​​\text{n}-1},$ here, $\text{n}=\frac27$

$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}=\frac{2}{7}(\text{a})^{\frac{2}{7}-1}$

$=\frac{2}{7}\text{a}^{\frac{-5}{7}}$ 

$=\lim\limits_{\text{x}\rightarrow1}\frac{\big(\sqrt{5\text{x}-4}-\sqrt{\text{x}}\big)}{(\text{x}-1)\big(\text{x}^2+1+1\big)}\times\frac{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}{\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{(5\text{x}-4-\text{x})}{(\text{x}-1)\big(\text{x}^2+1+\text{x}\big)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{4(\text{x}-1)}{(\text{x}-1)\big(\text{x}^2+\text{x}+1\big)\big(\sqrt{5\text{x}-4}+\sqrt{\text{x}}\big)}$

$=\frac{4}{(1+1+1)\big(\sqrt{5-4}+\sqrt{1}\big)}$

$=\frac{4}{(3)(1+1)}$

$=\frac{4}{3\times2}=\frac23$ 

$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{3}}\frac{\text{x}^{\text{n}}-3^{\text{n}}}{\text{x}-3}$

Applying the formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$

Here, n = n, a = 3

$\Rightarrow{\lim\limits_{\text{x}\rightarrow{3}}}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-3}=\text{n}(3)^{\text{n}-1}$

It is given that $\text{n}(3)^{\text{n}-1}=108$

$\Rightarrow\text{n}(3)^{\text{n}-1}=2\times2\times3\times3\times3$

$=(2)^2\times(3)^3$

$=4(3)^{4-1}$

$\Rightarrow\text{n}=4$ 

$=\lim\limits_{\text{h}\rightarrow0}\frac{\big(\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}}\big)}{\text{h}}\times\frac{\big(\sqrt{\text{x}+\text{h}}+\sqrt{\text{x}}\big)}{\big(\sqrt{\text{x}+\text{h}}+\sqrt{\text{x}}\big)}$

$=\lim\limits_{\text{h}\rightarrow0}\frac{(\text{x}+\text{h}-\text{x})}{\text{h}\big(\sqrt{\text{x}+\text{h}}+\sqrt{\text{x}}\big)}$

$=\frac{1}{\sqrt{\text{x}}+\sqrt{\text{x}}}$

$=\frac{1}{2\sqrt{\text{x}}}$ 

$=\lim\limits_{\text{x}\rightarrow0}\frac{\frac{(1+\text{x})^{{{6}}}-1^6}{1+\text{x}-1}}{\frac{(1+\text{x})^2-1^2}{1+\text{x}-1}}$

$\Rightarrow \text{Let} 1 + \text{x} = \text{y}, \text{as x} → 0, \text{y} → 1$

$=\frac{\lim\limits_{\text{y}\rightarrow1}\frac{\text{y}^6-1^6}{\text{y}-1}}{\lim\limits_{\text{y}\rightarrow1}\frac{\text{y}^2-1}{\text{y}-1}}$

$=\frac{6(1)^{6-1}}{2(1)^{2-1}}$ $\Big[\text{Using formula} \lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{a}-\text{a}}=\text{na}^{\text{n}-1}\Big]$

$=\frac62$

$=3$ 

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}^2}-\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^2}-\sqrt{1+\text{x}}\big)}\times\frac{\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^3}-\sqrt{1+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(1+\text{x}^2\big)-\big(1+\text{x}\big)\times\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^3}-\sqrt{1+\text{x}}\big)\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{x}^2-\text{x}\big)\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)\big(1+\text{x}^3-1-\text{x}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}(\text{x}-1)\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)\times\big(\text{x}^2-1\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}(\text{x}-1)\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}{\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)(\text{x})(\text{x}-1)(\text{x}+1)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}^3}+\sqrt{1+\text{x}}\big)}{\big(\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}}\big)(\text{x}+1)\big)}$

$=\frac{2}{2}=1$

$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}+1)\big(\text{x}^2+4\text{x}-2\big)}{(\text{x}-1)\big(\text{x}^2-4\text{x}+1\big)}$

$=\frac{(1)^24(1)-2}{(1)^24(1)+1}$

$=\frac{1+4-2}{1+4+1}$

$=\frac{3}{6}$

$=\frac{1}{2}$

$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$ [Dividing numrator and denominator by x - a]

$=\frac{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$

Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$

Here, $\text{n}=\frac57$ is numerator and applying $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^\text{m}-\text{a}^\text{m}}{\text{x}-\text{a}}=\text{ma}^{\text{m}-1}$ in denominator, where $\text{m}=\frac27$

$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}=\frac{\frac{5}{7}\text{a}^{\frac{5}{7}-1}}{\frac{2}{7}(\text{a})^{\frac{2}{7}-1}}$

$=\frac{\frac{5}{7}\text{a}^{\frac{-2}{7}}}{\frac{2}{7}\text{a}^{\frac{-5}{7}}}$

$=\frac{5}{2}\text{a}^{\frac{-2}{7}+\frac{5}{7}}$

$=\frac{5}{2}\text{a}^{\frac{3}{7}}$ 

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{\sqrt{5+2\text{x}}-\Big(\sqrt{\big(\sqrt{3}+\sqrt{2}\big)^2}\Big)}{\text{x}^2-6}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{5+2\sqrt{6}}\big)}{\text{x}^2-6}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{5+2\sqrt{6}}\big)}{\text{x}^2-6}\times\frac{\sqrt{5+2\text{x}}+\big(\sqrt{5+2\sqrt{6}}\big)}{\sqrt{5+2\text{x}}+\big(\sqrt{5+2\sqrt{6}}\big)}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{10}}}\frac{{7+2\text{x}}-{7-2\sqrt{10}}}{\big(\text{x}^2-10\big)\Big(\sqrt{7+2\text{x}}+\big(\sqrt{7+2\sqrt{10}}\big)\Big)}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{5+2\text{x}-5-2\sqrt{6}}{\big(\text{x}^2-6\big)\Big(\sqrt{5+2\text{x}}+\big(\sqrt{5+2\sqrt{6}}\big)\Big)}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{2\big(\text{x}-\sqrt{6}\big)}{\big(\text{x}^2-\sqrt{6}\big)\Big(\sqrt{5+2\text{x}}+\big(\sqrt{5+2\sqrt{10}}\big)\Big)}$

$=\lim\limits_{\text{x}\rightarrow{\sqrt{6}}}\frac{2}{\big(\text{x}+\sqrt{6}\big)\Big(\sqrt{5+2\text{x}}+\big(\sqrt{5+2\sqrt{6}}\big)\Big)}$

$=\frac{2}{\big(2\sqrt{6}\big)\Big(2\sqrt{5+2\sqrt{6}}\Big)}$

$=\frac{1}{\big(2\sqrt{6}\big)\Big(\sqrt{5+2\sqrt{6}}\Big)}$

$=\frac{1}{\big(2\sqrt{6}\big)\Big(\sqrt{3}+\sqrt{2}\Big)}$

$=\frac{\big(\sqrt{3}-\sqrt{2}\big)}{\big(2\sqrt{6}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}^2}-\sqrt{1-\text{x}^2}\big)}{\text{x}}\times\frac{\big(\sqrt{1+\text{x}^2}+\sqrt{1-\text{x}^2}\big)}{\big(\sqrt{1+\text{x}^2}+\sqrt{1-\text{x}^2}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(1+\text{x}^2\big)-\big(1-\text{x}^2\big)}{\text{x}\big(\sqrt{1+\text{x}^2}+\sqrt{1-\text{x}^2}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}^2}{\text{x}\big(\sqrt{1+\text{x}^2}+\sqrt{1+\text{x}^2}\big)}$

$=\frac{2\times0}{\big(\sqrt{1}+\sqrt{1}\big)}$

$=\frac{2}{2}\times0$

$=0$ 

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}{\text{x}\times\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{(2-\text{x})-(2+\text{x})}{\text{x}\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{-2\text{x}}{\text{x}\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$

$=\frac{-2}{\sqrt{2}+\sqrt{2}}$

$=\frac{-2}{2\sqrt{2}}$

$=\frac{-1}{\sqrt{2}}$

Now $\text{x}^2-\text{x}-6$

$=\text{x}^2-3\text{x}+2\text{x}-6$

$=\text{x}(\text{x}-3)+2(\text{x}-3)$

$=(\text{x}+2)(\text{x}-3)\ \cdots(\text{i})$

Dividing $\text{x}^3-3\text{x}^2+\text{x}-3\text{ by }(\text{x}-3), \text{ we get}$ 

Thus (x - 3) is a factor of $\text{x}^3-3\text{x}^2+\text{x}-3\ \cdots(\text{ii})$

Substituting (i) and (ii) in the given expression

$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}+2)(\text{x}-3)}{\big(\text{x}^2+1\big)(\text{x}-3)}$

$=\frac{\text{x}+2}{\text{x}^2+1}=\frac{3+2}{9+1}=\frac{5}{10}$

$=\frac12$

$=\lim\limits_{\text{x}\rightarrow2}\Bigg\{\frac{1}{\text{x}-2}-\frac{2(2\text{x}-3)}{\text{x}\big(\text{x}^2-2\text{x}-\text{x}+2\big)}\Bigg\}$

$= \lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{1}{\text{x}-2}-\frac{2(2\text{x}-3)}{\text{x}(\text{x}-2)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{\text{x}(\text{x}-1)-2(2\text{x}-3)}{\text{x}(\text{x}-1)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{\text{x}^2-\text{x}-4\text{x}+6}{\text{x}(\text{x}-1)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{\text{x}^2-2\text{x}-3\text{x}+6}{\text{x}(\text{x}-1)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{\text{x}(\text{x}-2)-3(\text{x}-2)}{\text{x}(\text{x}-1)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{(\text{x}-2)(\text{x}-3)}{\text{x}(\text{x}-1)(\text{x}-2)}\bigg\}$

$=\lim\limits_{\text{x}\rightarrow2}\bigg\{\frac{1}{\text{x}-2}\frac{2(2\text{x}-3)}{\text{x}(\text{x}-2)(\text{x}-1)}\bigg\}$

$=\frac{-1}{2}$ 

$=\lim\limits_{\text{x}\rightarrow{-1}}\frac{\text{x}^3-(-1)^3}{\text{x}-(-1)}$ [Dividing numerator and denominator by x - 1]

Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$

Here, n = 3, a = -1

$\Rightarrow\lim\limits_{\text{x}\rightarrow-1}\frac{\text{x}^3-(-1)^3}{\text{x}-(-1)}=\text{na}^{\text{n}-1}$

$=3(-1)^{3-1}$

$=3(-1)^2$

$=3$ 

$\Rightarrow\text{x}\rightarrow\frac{\pi}{4},$ then $\frac\pi4-\text{x}\rightarrow0$ let $\frac\pi4-\text{x}=\text{y}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\cos\big(\frac\pi4+\text{y}\big)-\sin\big(\frac\pi4+\text{y}\big)}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\Big[\big(\cos\frac\pi4\cos\text{y}-\sin\frac\pi4\sin\text{y}\big)-\big(\sin\frac\pi4\cos\text{y}+\cos\frac\pi4\sin\text{y}\big)\Big]}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\Big[\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}-\frac{\cos\text{y}}{\sqrt{2}}-\frac{\sin\text{y}}{\sqrt{2}}\Big]}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$

$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\frac{-2\sin\text{y}}{\sqrt{2}}}{-\text{y}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$

$=\sqrt{2}\lim\limits_{\text{y}\rightarrow{0}}\Big(\frac{\sin\text{y}}{\text{y}}\Big)\times\frac{1}{\lim\limits_{\text{y}\rightarrow{0}}\big(\cos\big(\frac\pi4+\text{y}\big)+\sin\big(\frac\pi4+\text{y}\big)\big)}$

$=\sqrt{2}\times1\times\frac{1}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=\sqrt{2}\times\frac{1}{\frac{2}{\sqrt{2}}}$

$=\frac{\sqrt{2}\times\sqrt{2}}{2}=1$ 

$=\lim\limits_{\text{x}\rightarrow\infty}\Big(\sqrt{\text{x}^2+\text{x}}-{\text{x}}\Big)$

$=\lim\limits_{\text{n}\rightarrow\infty}\Bigg(\big(\sqrt{\text{x}^2+\text{x}}-\text{x}\big)\times\frac{\big(\sqrt{\text{x}^2+\text{x}}+\text{x}\big)}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\Bigg)$

$=\lim\limits_{\text{n}\rightarrow\infty}\Bigg(\frac{\big(\text{x}^2+\text{x}\big)-\text{x}^2}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\Bigg)$

$=\lim\limits_{\text{n}\rightarrow\infty}\bigg(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}}+\text{x}}\bigg)$ $\Big[\frac{\infty}{\infty}\text{ from}\Big]$

$=\lim\limits_{\text{n}\rightarrow{\infty}}\begin{pmatrix}\frac{1}{\sqrt{\frac{\text{x}^2}{\text{x}^2}}+\frac{\text{x}}{\text{x}^2}+1}\end{pmatrix}$

$=\lim\limits_{\text{n}\rightarrow{\infty}}\begin{pmatrix}\frac{1}{\sqrt{1+\frac{1}{\text{x}}}+1}\end{pmatrix}$

$=\frac{1}{1+1}$

$=\frac12$

$=\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}-3}{\big(\sqrt{\text{x}-2}-\sqrt{4-\text{x}}\big)}\times\frac{\big(\sqrt{\text{x}}-2+\sqrt{4-\text{x}}\big)}{\big(\sqrt{\text{x}}-2+\sqrt{4-\text{x}}\big)}$

$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}-3)\big(\sqrt{\text{x}-2}+\sqrt{4-\text{x}}\big)}{(\text{x}-2)-(4-\text{x})}$

$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}-3)\big(\sqrt{\text{x}-2}+\sqrt{4-\text{x}}\big)}{\text{x}-2-4+\text{x}}$

$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}-3)\big(\sqrt{\text{x}-2}+\sqrt{4-\text{x}}\big)}{2(\text{x}-3)}$

$=\frac{1}{2}\lim\limits_{\text{x}\rightarrow3}\big(\sqrt{\text{x}-2}+\sqrt{4-\text{x}}\big)$

$=\frac{1}{2}\big(\sqrt{3-2}+\sqrt{4-\text{x}}\big)$

$=\frac{1}{2}\big(\sqrt{1}+\sqrt{1}\big)$

$=\frac{1}{2}(1+1)=\frac22$

$=1$ 

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}}\big)}{\text{x}\sqrt{\text{a}^2+\text{ax}}}\times\frac{\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}{\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{(\text{a}+\text{x})-\text{a}}{\text{x}\sqrt{\text{a}^2+\text{ax}}\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{\text{x}\sqrt{\text{a}^2+\text{ax}}\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\big(\sqrt{\text{a}^2+\text{ax}}\big)\big(\sqrt{\text{a}+\text{x}}+\sqrt{\text{a}}\big)}$

$=\frac{1}{\text{a}\big(\sqrt{2\text{a}}\big)}$

$=\frac{1}{2\text{a}\sqrt{\text{a}}}$ 

$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\times\big(\sqrt{\text{x}^2+3}+2\big)}{\big(\sqrt{\text{x}^2+3}-2\big)\big(\sqrt{\text{x}^2+3}+2\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\big(\sqrt{\text{x}^2+3}+2\big)}{\big({\text{x}^2+3}-4\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{(\text{x}-1)\big(\sqrt{\text{x}^2+3}+2\big)}{\big({\text{x}^2-1}\big)}$

$=\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{\text{x}^2+3}+2}{{\text{x}}+1}$

Putting the value x = 1

$\Rightarrow\frac{\sqrt{1+3}+2}{1+1}$

$=\frac{2+2}{2}$

$=\frac{4}{2}=2$