Limits and Cont...

The graph of the function $$y = f (x)$$ has a unique tangent at the point $$(e^{a} ,0)$$ through which the graph passes then $$\displaystyle \lim_{x\rightarrow e^{a}}\frac{log_{e}\{1+7f(x)\}-sinf(x)}{3f(x)}$$ is

$$\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} \frac { \cos ^{ 2 }{ x } -\sin ^{ 2 }{ x } -1 }{ \sqrt { { x }^{ 2 }+4 } -2} , & x\neq 0 \end{matrix} \\ \begin{matrix} a, & x=0 \end{matrix} \end{cases}$$ then the value of $$a$$ in order that $$f(x)$$ may be continuous at $$x=0$$ is

If $$\displaystyle f(x)=\frac{\sin 3x+A\sin 2x+B\sin x}{x^{5}};x\neq 0$$ is continuous at $$x=0$$ , then

$$f(x)=\left\{\begin{array}{ll}\dfrac{(x+bx^{2})^{1/_{2}}-x^{1/2}}{bx^{3/2}} & x>0\\c & x=0\\\dfrac{\sin(a+1)x+\sin x}{x} & x<0\end{array}\right.$$ is continuous at $${x}=0$$, then

Assertion(A): $$f(x)=\left\{\begin{array}{ll}x^{2}\sin(\frac{1}{x}) , & x\neq 0\\0, & x=0\end{array}\right.$$ is continuous at $${x}=0$$
Reason(R): Both $$h(x)=x^{2},g(x)=
\left\{\begin{array}{ll}\sin(\frac{1}{x}) , & x\neq 0\\0, & x=0\end{array}\right.$$are continuous at $$x = 0$$

If $$\phi (x) =\displaystyle \lim_{n \rightarrow \infty} \frac{x^{2n} f(x) + g(x)}{1 + x^{2n}}$$, then

If $${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },..,{ x }_{ n }$$ are the roots of the equation $$x^n+ax+b=0,$$ then the value of $$\left( { x }_{ 1 }-{ x }_{ 2 } \right) \left( { x }_{ 1 }-{ x }_{ 3 } \right) \left( { x }_{ 1 }-{ x }_{ 4 } \right) ...\left( { x }_{ 1 }-{ x }_{ n } \right) $$ is equal to

If $$\displaystyle f(x) = \frac{x - e^x + cos  2x}{x^2}, x \neq 0$$, is continuous at $$x = 0$$, 
where [x] and {x} denotes the greatest integer and fractional part functions, respectively.

If $$f(x) = \displaystyle \left\{\begin{matrix}\dfrac{8^x - 4^x - 2^x+1}{x^2}, & x>0\\ e^x \sin  x+ \pi  x + \lambda  \ln  4, & x \leq 0\end{matrix}\right.$$ is continuous at $$x = 0$$, then the value of $$\lambda$$ is

STATEMENT-1 : $$\displaystyle \lim_{x \rightarrow 0} [x] \left \{ \frac{e^{1/x} - 1}{e^{1/x} + 1} \right \}$$ (where [.] represents the greatest integer function) does not exist.
STATEMENT-2 : $$\displaystyle \lim_{x \rightarrow 0} \left ( \frac{e^{1/x} - 1}{e^{1/x} + 1} \right )$$ does not exists.