Limits, Continuity and Differentiability

\(\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}\) is equal to

Consider the function, \(f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R\)
Statement - 1 : \(f'\left( 4 \right) = 0\)
Statement - 2 : \(f\) is continuous in [2, 5], differentiable in (2, 5) and \(f\)(2) = \(f\)(5...

If \(f:R \to R\) is a function defined by
\(f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi\),
where [x] denotes the greatest integer function, then \(f\) is

\(\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)\)

The value of \(p\) and \(q\) for which the function
$$f\left( x \right) = \left\{ {\matrix{
{{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr
q & {,x = 0} \cr
{{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} ...

Let \(f:R \to R\) be a positive increasing function with
\(\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1\). Then \(\mathop {\lim }\limits_{x \to \infty } {{f(2x)} \over {f(x)}} =\)

Let \(f\left( x \right) = x\left| x \right|\) and \(g\left( x \right) = \sin x.\)
Statement-1: gof is differentiable at \(x=0\) and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at \(x=0\).

Let \(f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right.\)
Then which one of the following is true?

The function \(f:R/\left\{ 0 \right\} \to R\) given by
\(f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}\)
can be made continuous at \(x\) = 0 by defining \(f\)(0) as

Let \(f:R \to R\) be a function defined by
\(f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}\), then which of the following is true?

The set of points where \(f\left( x \right) = {x \over {1 + \left| x \right|}}\) is differentiable is

If \(f\) is a real valued differentiable function satisfying
\(\left| {f\left( x \right) - f\left( y \right)} \right|\) \(\le {\left( {x - y} \right)^2}\), \(x, y\) \(\in R\)
and \(f(0)\) = 0, then \(f(1)\) equals

Suppose \(f(x)\) is differentiable at x = 1 and
\(\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5\), then \(f'\left( 1 \right)\) equals

Let \(\alpha\) and \(\beta\) be the distinct roots of \(a{x^2} + bx + c = 0\), then
\(\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}\) is equal to

If \(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}\), then the value of \(a\) and \(b\), are

Let \(f(x) = {{1 - \tan x} \over {4x - \pi }}\), \(x \ne {\pi \over 4}\), \(x \in \left[ {0,{\pi \over 2}} \right]\).
If \(f(x)\) is continuous in \(\left[ {0,{\pi \over 2}} \right]\), then \(f\left( {{\pi \over 4}} \right)\) is

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

If \(\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}\) = k, the value of k is

If \(f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \cr } } \right.\)
then \(f(x)\) is

Let \(f(a) = g(a) = k\) and their nth derivatives
\({f^n}(a)\), \({g^n}(a)\) exist and are not equal for some n. Further if
\(\mathop {\lim }\limits_{x \to a} {{f(a)g(x) - f(a) - g(a)f(x) + f(a)} \over {g(x) - f(x)}} = 4\)
then the value o...

\(f\) is defined in \(\left[ { - 5,5} \right]\) as
\(f\left( x \right) = x\) if \(x\) is rational
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\) \(= - x\) if \(x\) is irrational. Then

Let \(f(2) = 4\) and \(f'(x) = 4.\)
Then \(\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \over {x - 2}}\) is given by

f(x) and g(x) are two differentiable functions on [0, 2] such that
f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9
then f(x) - g(x) at x = \({3 \over 2}\) is

\(\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}\)

\(\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}\), \(n \in N\), ( [x] denotes the greatest integer less than or equal to x )

\(\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}\) is

If \(f\left( 1 \right) = 1,{f'}\left( 1 \right) = 2,\) then
\(\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}}\) is

If f(x + y) = f(x).f(y) \(\forall\) x, y and f(5) = 2, f'(0) = 3, then
f'(5) is