\(\lim_{ h\rightarrow 0 } \frac { (a+h)^{ 2 }sin(a+h)-a^{ 2 }sin\quad a }{ h } \)
\(=\lim_ { lim }{ h\rightarrow 0 } \frac { (a^{ 2 }+h^{ 2 }+2ah)[sin \ a\ cos \ h+cos\ a\ sin \ h)-a^{ 2 }sin\quad a }{ h } \)
\(\left[ \because \ sin(C+D)=sinCcosD+CosCsinD \right] \)
\(=\lim_{ h\rightarrow 0 } \left[ \frac { a^{ 2 }sin \ a(cos \ h-1) }{ h } +\frac { a^{ 2 }cos \ a \ sin \ h) }{ h } +(h+2a)(sin \ a \ cos \ h+cos\ a \ sin\ h) \right] \)
\(=\lim_{ h\rightarrow 0 } \left[ \frac { a^{ 2 }sina(-2sin^{ 2 }\frac { h }{ 2 } ) }{ \frac { h^{ 2 } }{ 2 } } .\frac { h }{ 2 } \right] +\overset { lim }{ h\rightarrow 0 } \frac { a^{ 2 }cosasinh }{ h } +\lim_{ h\rightarrow 0 } (h+2a)sin(a+h)\)
\(\left[ \because cosm \ h-1=-2sin^{ 2 } h/2\quad and\quad sinacos\quad h+cosasin\quad h=sin(a+h) \right] \)
\(=a^{ 2 }sin\quad a\times 0+a^{ 2 }\quad cosa(1)+2asina \quad \left[ \because \lim_{ x\rightarrow 0 } \frac { sin\quad x }{ x } =1 \right]\)
\( =a^{ 2 }cos \ a+2asin \ a\)

\( \lim _{x \rightarrow \pi / 6} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2} \)
\(= \lim _{x \rightarrow \pi / 6} \frac{\operatorname{cosec}^{2} x-1-3}{\operatorname{cosec} x-2} \quad\left[\because \operatorname{cosec}^{2} x-\cot ^{2} x=1\right] \)
\(= \lim _{x \rightarrow \pi / 6} \frac{\operatorname{cosec}^{2} x-4}{\operatorname{cosec} x-2} \)
\(= \lim _{x \rightarrow \pi / 6} \frac{(\operatorname{cosec} x-2)(\operatorname{cosec} x+2)}{(\operatorname{cosec} x-2)} \quad[\text { by factorisation }] \)
\(=\lim _{x \rightarrow \pi / 6}(\operatorname{cosec} x+2)\)
\(= \operatorname{cosec} \frac{\pi}{6}+2=2+2=4\)

\(\lim _{x \rightarrow 2} \frac{x^{2}-x \log x+2 \log x-4}{x-2} \)
\(=\lim _{x \rightarrow 2} \frac{\left(x^{2}-4\right)-\log x(x-2)}{x-2} \)
\(=\lim _{x \rightarrow 2} \frac{(x+2)(x-2)-\log x(x-2)}{(x-2)} \)
\(=\lim _{x \rightarrow 2} \frac{(x-2)[x+2-\log x]}{(x-2)} \)
\(=\lim _{x \rightarrow 2}[x+2-\log x]=2+2-\log 2 \)
\(=4-\log 2\)

\(\lim_ { x\rightarrow 0 }{ lim } \frac { x\quad tan4x }{ 1-cos4x } =\lim_ { x\rightarrow 0 }{ lim } \frac { x.\frac { sin\quad 4x }{ cos\quad 4x } }{ 1-cos\quad 2.(2x) } \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { x.\frac { sin\quad 4x }{ cos\quad 4x } }{ 1-\left( 1-2{ sin }^{ 2 }2x \right) } \quad [\because \quad cos2\theta =1-2{ sin }^{ 2 }]\)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { x.sin\quad 4x }{ cos\quad 4x.\quad 2\quad { sin }^{ 2 }2x } \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { x.sin\quad 2(2x) }{ cos\quad 4x(2.sin\quad 2x.sin\quad 2x) } \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { x\quad (2\quad sin\quad 2x.cos\quad 2x) }{ (2\quad sin\quad 2x.sin\quad 2x)cos\quad 4x } \quad [\because \quad sin\ 2\theta =2\quad sin\quad \theta \quad cos\quad \theta ]\)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { x.\quad cos\quad 2x }{ (sin\quad 2x).cos\quad 4x } =\lim_ { x\rightarrow 0 }{ lim } \frac { cos\quad 2x }{ cos\quad 4x } .\left( \frac { x }{ sin\quad 2x } \right) \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { 1 }{ \left( \frac { sin\quad 2x }{ x } \times \frac { 2 }{ 2 } \right) } \times \lim_ { x\rightarrow 0 }{ lim } \frac { cos\quad 2x }{ cos\quad 4x } \)
\(=\frac { 1 }{ 2 } \lim_ { x\rightarrow 0 }{ lim } \frac { 1 }{ \left( \frac { sin\quad 2x }{ 2x } \right) } \times \lim_ { x\rightarrow 0 }{ lim } \frac { cos\quad 2x }{ cos\quad 4x } \)
\(=\frac { 1 }{ 2 } \times \frac { 1 }{ 1 } \times \frac { cos\quad 0 }{ cos\quad 0 } =\frac { 1 }{ 2 } \times \frac { 1 }{ 1 } \times \frac { 1 }{ 1 } =\frac { 1 }{ 2 } \quad \left[ \because \lim_{ x\rightarrow 0 }{ lim } \frac { sin\quad \theta }{ \theta } \quad =1 \right] \)

\(\lim_ { x\rightarrow 0 }{ lim } \frac { sin\quad x-2sin\quad 3x+sin\quad 5x }{ x } \)
\(=\lim_{ x\rightarrow 0 }{ lim } \frac { \left( sin5x\quad +\quad sin\quad x \right) -2sin\quad 3x }{ x } \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { 2sin\left( \frac { 5x+x }{ 2 } \right) cos\left( \frac { 5x-x }{ 2 } \right) -2sin3x }{ x } \)
\( \left[ \because \ sin\ C+sin\ D=2sin\ \left( \frac { C+D }{ 2 } \right) COS\left( \frac { C-D }{ 2 } \right) \right] \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { 2sin3x\quad cos2x-2sin3x }{ x } \)
\(=\lim_ { x\rightarrow 0 }{ lim } \frac { 2sin3x(cos\quad 2x-1) }{ x } \)
\(=\lim_ { x\rightarrow 0 }{ lim } 2\left( \frac { sin\quad 3x }{ 3x } \right) \times 3\left[ \frac { -1(1-cos2x) }{ 1 } \right] \)
\(=-2\times 1\times 3(1-cos\quad 0) \quad \left[ \because \quad \underset { x\rightarrow 0 }{ lim } \frac { sin\quad \theta }{ \theta } \right] \)
\(=-6(1-1)=-6\times 0=0\)