The value of \mathop {\lim }\limits_{n \to \infty } 6\tan \left\{ {\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {{1 \over {{r^2} + 3r + 3}}} \right)} } \right\} is equal to :

Let f : R $$\to$$ R be defined as f(x) = \left[ {\matrix{ {[{e^x}],} & {x where a, b, c\in$$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

Let a be an integer such that \mathop {\lim }\limits_{x \to 7} {{18 - [1 - x]} \over {[x - 3a]}} exists, where [t] is greatest integer $$\le$$ t. Then a is equal to :

\mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}} is equal to :

Let f, g : R $$\to$$ R be two real valued functions defined as f(x) = \left\{ {\matrix{ { - |x + 3|} & , & {x 1 and k2 are real constants. If (gof) is differentiable at x = 0, then (gof) (-$$ 4) + (gof) (4) is equal to :

\mathop {\lim }\limits_{x \to 0} {{\cos (\sin x) - \cos x} \over {{x^4}}} is equal to :

Let f(x) = min {1, 1 + x sin x}, 0 $$\le$$ x $$\le$$ 2$$\pi$$. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right) is equal to

Let f(x) be a polynomial function such that $$f(x)+f^{\prime }(x)+f^{\prime \prime }(x)=x^5+64$$. Then, the value of \mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}} is equal to:

Let f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| where [t] denotes greatest integer\le$$t.Ifmisthenumberofpointswhere$$f$$isnotcontinuousandnisthenumberofpointswhere$$f$$ is not differentiable, then the ordered pair (m, n) is :

If $$\underset{n\to \infty }{\lim }\left(\sqrt{n^2-n-1}+n\alpha +\beta \right)=0$$, then $$8(\alpha +\beta )$$ is equal to :

$$\underset{x\to \frac{\pi }{4}}{\lim }\frac{8\sqrt{2}-(\cos x+\sin x{)}^7}{\sqrt{2}-\sqrt{2}\sin 2x}$$ is equal to

Let f : R $$\to$$ R be a continuous function such that $$f(3x)-f(x)=x$$. If $$f(8)=7$$, then $$f(14)$$ is equal to :

If the function f(x) = \left\{ {\matrix{ {{{{{\log }_e}(1 - x + {x^2}) + {{\log }_e}(1 + x + {x^2})} \over {\sec x - \cos x}}} & , & {x \in \left( {{{ - \pi } \over 2},{\pi \over 2}} \right) - \{ 0\} } \cr k & , & {x = 0} \cr } } \right. is continuous at x = 0, then k is equal to:

If f(x) = \left\{ {\matrix{ {x + a} & , & {x \le 0} \cr {|x - 4|} & , & {x > 0} \cr } } \right. and $$g(x) = \left\{ {\matrix{ {x + 1} & , & {x

Let f(x) = \left\{ {\matrix{ {{x^3} - {x^2} + 10x - 7,} & {x \le 1} \cr { - 2x + {{\log }_2}({b^2} - 4),} & {x > 1} \cr } } \right.. Then the set of all values of b, for which f(x) has maximum value at x = 1, is :

Let $$\beta =\underset{x\to 0}{\lim }\frac{\alpha x-\left(e^{3x}-1\right)}{\alpha x\left(e^{3x}-1\right)}$$ for some $$\alpha \in \mathbb{R}$$. Then the value of $$\alpha +\beta$$ is :

If for $$\mathrm{p}\ne \mathrm{q}\ne 0$$, the function $$f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q}x}-9}$$ is continuous at $$x=0$$, then :

The function $$f:\mathbb{R}\to \mathbb{R}$$ defined by $$f(x)=\underset{n\to \infty }{\lim }\frac{\cos (2\pi x)-x^{2n}\sin (x-1)}{1+x^{2n+1}-x^{2n}}$$ is continuous for all x in :

If $$\underset{x\to 0}{\lim }\frac{\alpha {\mathrm{e}}^x+\beta {\mathrm{e}}^{-x}+\gamma \sin x}{x{\sin }^{2}x}=\frac{2}{3}$$, where $$\alpha ,\beta ,\gamma \in \mathbf{R}$$, then which of the following is NOT correct?