Question Text
Question 2 :
If $f(x)=\left\{\begin{matrix}<br>4x, & x < 0\\ <br>1, & x=0\\<br>3x^2, & x > 0<br>\end{matrix}\right.$ then $\displaystyle \lim_{x\rightarrow 0}f(x)$ equals<br>
Question 4 :
$ \lim _{ x\rightarrow 1 }{ \dfrac { { { x }^{ n }-1 }  }{ x-1 }  }$ is equal to
Question 6 :
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 8 :
If $p\left( x \right) ={ a }_{ 0 }+{ a }_{ 1 }x+...+{ a }_{ n }{ x }^{ n }$ and $\left| p\left( x \right)  \right| \le \left| { e }^{ x-1 }-1 \right| $ for all $x\ge 0,$ then $\left| { a }_{ 1 }+2{ a }_{ 2 }+3{ a }_{ 3 }+...+n{ a }_{ n } \right| $
Question 10 :
<p>Find the left and right hand limits of $f(x)=\left\{\begin {array}\\ \dfrac{3x^2+2}{3x-2} \quad x<1 \\\dfrac {4x^2-3}{4x+3} \quad x>1\end {array}\right.$ at $x=1$</p>
Question 11 :
Let $f(x) \displaystyle = \frac{x^2 - 9x + 20}{x -[x]}$ where [x] is the greatest integer not greater than $x$, then
Question 12 :
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 13 :
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 15 :
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