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question_answer1) Find the value of \[\underset{x\to 0}{\mathop \lim }\,\left( \left[ \frac{100x}{\sin \,\,x} \right]+\left[ \frac{99\sin \,\,x}{x} \right] \right)\] (whrere \[[.]\] represents the greatest integral function)
question_answer2) Evaluate: \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+\frac{9}{{{n}^{3}}+1}+.....+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right].\]
question_answer3) If: \[\underset{x\to 0}{\mathop{\lim }}\,\frac{8}{{{x}^{8}}}\left[ 1-\cos \frac{{{x}^{2}}}{2}-\cos \frac{{{x}^{2}}}{4}+\cos \frac{{{x}^{2}}}{2}\cos \frac{{{x}^{2}}}{4} \right]\] is equal to \[\frac{\lambda }{k}\] then find \[k-\lambda \].
question_answer4) Evaluate: \[\underset{x\to \frac{\pi }{6}}{\mathop{\lim }}\,\frac{2{{\sin }^{2}}x+\sin \,x-1}{2{{\sin }^{2}}x-3\sin x+1}.\]
question_answer5) Evaluate: \[\underset{n\to \infty }{\mathop{\lim }}\,{{\left( {{3}^{n}}+{{4}^{n}} \right)}^{1/n}}.\]
question_answer6) If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{x\left( {{2}^{x}}-1 \right)}=k\,{{\log }_{2}}e\] then find k.
question_answer7) Evaluate: \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \left( {{\min }^{m}}\left( {{y}^{2}}-4y+11 \right) \right)\frac{\sin \,x}{x} \right]\] (where [ ] represent greatest integer function).
question_answer8) If \[f(x)=\frac{\sin ({{e}^{x-2}}-1)}{\log (x-1)},\] then find \[\underset{x\to 2}{\mathop{\lim }}\,f\left( x \right)\].
question_answer9) Evaluate:\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 4x}{1-\sqrt{\left( 1-x \right)}}\].
question_answer10) If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\left( 1+x \right)}^{1/x}}-e+\frac{1}{2}ex}{{{x}^{2}}}=\frac{11e}{k}\]then find k.
question_answer11) Let\[f(n)=\underset{x\to 0}{\mathop{\lim }}\,{{\left( \left( 1+\sin \frac{x}{2} \right)\left( 1+\sin \frac{x}{{{2}^{2}}} \right)...\left( 1+\sin \frac{x}{{{2}^{n}}} \right) \right)}^{1/x}}\]. If \[=\underset{n\to \infty }{\mathop{\lim }}\,f\left( n \right)=\frac{3e}{k}\]then find k.
question_answer12) If \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+.......+\frac{1}{\left( 4n-1 \right)\left( 4n+3 \right)} \right]=\frac{2}{k}\]then find k.
question_answer13) Let\[f\left( x \right)=3{{x}^{10}}-7{{x}^{8}}+5{{x}^{6}}-21{{x}^{3}}+3{{x}^{2}}-7\]. Then find the value of \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( 1-h \right)-f\left( 1 \right)}{{{h}^{3}}+3h}.\]
question_answer14) Evaluate: \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1+\cos \,\pi x}{{{\tan }^{2}}\pi x}\]
question_answer15) Evaluate: \[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{{{e}^{x}}-{{e}^{\sin x}}}{x-\sin x} \right)\]
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