Question 3 :
Find the value of k so that the function f is continuous at the indicated point.$f(x)={\begin{matrix} kx^2 & , x\leq 2 \\ 3 & , x>2 \end{matrix}}$ at $x=2$.
Question 4 :
The value of the limit $\displaystyle\lim _{ x\rightarrow 1 }{ \dfrac { \sin { \left( { e }^{ x-1 }-1 \right) } }{ \log { x } } } $ is
Question 6 :
$\underset { x\rightarrow 1 }{ Lt } { (1+\sin\pi x) }{ \pi x }$ 
Question 8 :
Identify the value of $\displaystyle\lim_{x \rightarrow 2} x^2 - 5x + 6$
Question 9 :
$ \lim _{ x\rightarrow 1 }{ \dfrac { { { x }^{ n }-1 }  }{ x-1 }  }$ is equal to
Question 11 :
If   ${z_r} = \cos \dfrac{{r\alpha }}{{{n^2}}} + i\sin \dfrac{{r\alpha }}{{{n^2}}}$, where $ r= 1, 2, 3, ....n$, then $\mathop {\lim }\limits_{n \to \infty } \left( {{z_1}.{z_2}.....{z_n}} \right)$ is equal to 
Question 12 :
The value of $\displaystyle \lim_{x\rightarrow \infty}\frac {2\sqrt x+3\sqrt [3]{x}+5\sqrt [5]{x}}{\sqrt {3x+2}+\sqrt [3]{2x-3}}=$<br>
Question 14 :
lf $f(x)=2x-3,a=2,l=1$ and $\epsilon =0.001$ then $\delta>0$ satisfying$ 0<|x-a|<\delta, \ \ |f(x)-l|<\epsilon$, is:<br/>
Question 16 :
Solve $\displaystyle \lim _{ x\rightarrow a }{ \dfrac { { x }^{ 2 }-\left( a+1 \right) x+a }{ { x }^{ 3 }-{ a }^{ 3 } }  } $
Question 17 :
Let $f(x)=\displaystyle \begin{cases} \dfrac{\sin[x]}{[x]}\>;\>[x] \neq0\\<br> 0 ; [x]=0\end{cases}$, then $\lim_{x\rightarrow 0}f(x)=$<br><br>
Question 18 :
If $A_{i}=\displaystyle \frac{x-a_{i}}{|x-a_{i}|}$ where $i=1,2,3$  and $a_{1}<a_{2}<a_{3}$ then $\displaystyle \lim_{x\rightarrow a_{2}}A_{1}A_{2}A_{3}=$<br/>
Question 19 :
If $\lim _{ x\rightarrow 0 }{ \left( { x }^{ -3 }\sin { 3x } +a{ x }^{ -2 }+b \right) } $ exists and is equal to $0$, then
Question 20 :
If $|x| < 1$, then $\displaystyle \lim_{n \rightarrow \infty }\{ (1 + x) (1+x^2)(1 + x^4) ..... (1 + x^{2n}) \}$ is equal to
Question 22 :
If $f\left( x+2 \right) =\dfrac { 1 }{ 2 } \left\{ f\left( x+1 \right) +\dfrac { 4 }{ f\left( x \right) } \right\} $ and $f\left( x \right) > 0$, for all $x\in R$, then $\displaystyle\lim _{ x\rightarrow \infty }{ f\left( x \right) } $ is
Question 24 :
$\displaystyle\lim _{ x\xrightarrow -1}\frac{1}{\sqrt{|x|-\{ -x \} }}$ (where { x }  denotes the fractional part of x) is equal to
Question 26 :
The value of $\displaystyle \lim _{ x\rightarrow a }{ \frac { \sqrt { x-b } -\sqrt { a-b } }{ { x }^{ 2 }-{ a }^{ 2 } } } \left( a>b \right) $ is
Question 27 :
If $\phi (x) =\displaystyle \lim_{n \rightarrow \infty} \frac{x^{2n} f(x) + g(x)}{1 + x^{2n}}$, then
Question 28 :
If $ \displaystyle \lim_{n \rightarrow \infty} \frac{(1^2 + 2^2 + .......+ n^2)(1^4 + 2^4 + .......+ n^4)}{(1^7 + 2^7+....+ n^7)} = \frac{K+1}{15};$ then K is equal to
Question 29 :
For every integer $n$, let $a_{n}$ and $b_{n}$ be real numbers. Let function $f: IR \rightarrow IR$ be given by<br>$f(x)=\left\{\begin{array}{l}<br>a_{n}+\sin\pi x,\ for\ x\in[2n,\ 2n+1]\\<br>b_{n}+\cos\pi x,\ for\ x\ \in(2n-1,2n) <br>\end{array}\right.$ <br> $, for\ all\ integers\ n.$<br>lf $f$ is continuous, then which of the following hold(s) for all $n$?<br>
Question 30 :
If $P=\lim _{ n\rightarrow \infty }{ \cfrac { { \left( \prod _{ r=1 }^{ n }{ \left( { n }^{ 3 }+{ r }^{ 3 } \right) } \right) }^{ 1/n } }{ { n }^{ 3 } } } $ and $\lambda =\int _{ 0 }^{ 1 }{ \cfrac { dx }{ 1+{ x }^{ 3 } } } $ then $\ln {P}$ is equal to
Question 34 :
Evaluate$lim_{x\to 0} \dfrac{\sqrt[K] {1+x} -1}{x}$ ( K is a positive integer )
Question 36 :
Let $f(x) \displaystyle = \frac{x^2 - 9x + 20}{x -[x]}$ where [x] is the greatest integer not greater than $x$, then
Question 37 :
Arrange the following limits in the ascending order :<br>(1) $\lim _ { x \rightarrow \infty } \left( \dfrac { 1 + x } { 2 + x } \right) ^ { x + 2 }$<br><br>(2) $\lim _ { x \rightarrow 0 } ( 1 + 2 x ) ^ { 3 / x }$<br><br>(3) $\lim _ { \theta \rightarrow 0 } \dfrac { \sin \theta } { 2 \theta }$<br><br>(4) $\lim _ { x \rightarrow 0 } \dfrac { \log _ { e } ( 1 + x ) } { x }$
Question 38 :
The value of $\displaystyle \lim_{x \rightarrow 1^{-}}\dfrac {1 - \sqrt {x}}{(\cos^{-1}x)^{2}}$
Question 39 :
$\lim _{ x\rightarrow { 0 }^{ + } }{ \left( { \left( x\cos { x }  \right)  }^{ x }+{ \left( \cos { x }  \right)  }^{ \frac { 1 }{ \ln { x }  }  }+{ \left( x\sin { x }  \right)  }^{ x } \right)  } $ is equal to<br/>
Question 40 :
$\displaystyle \lim_{x\to\infty}{\displaystyle \frac{2\sqrt{x}+3\sqrt [\Large 3]{x}+4\sqrt [\Large 4]{x}+...+n\sqrt [\Large n]{x}}{\sqrt{(2x-3)}+\sqrt[\Large 3]{(2x-3)}+\sqrt[\Large 4]{(2x-3)}+...+\sqrt[\Large n]{(2x-3)}}}$ is equal to