Limits, Continuity and Differentiability

If a function f(x) defined by
\(f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1 \le x \le 3} \cr {a{x^2} + 2cx,} & {3 < x \le 4} \cr } } \right.\)
be continuous for some $...

\(\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}\) is equal to :

\(\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}\)

Let f : R \(\to\) R be a function defined as
\(f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.\)
Then, f is

For each x\(\in\)R, let [x] be the greatest integer less than or equal to x.
Then \(\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}\) is e...

Let ƒ(x) = 15 – |x – 10|; x \(\in\) R. Then the set
of all values of x, at which the function,
g(x) = ƒ(ƒ(x)) is not differentiable, is :

If the function ƒ defined on , \(\left( {{\pi \over 6},{\pi \over 3}} \right)\) by
$$$f(x) = \left\{ {\matrix{
{{{\sqrt 2 {\mathop{\rm cosx}\nolimits} - 1} \over {\cot x - 1}},} & {x \ne {\pi \over 4}} \cr
{k,} & {x = {\pi \ove...

If \(f(x) = [x] - \left[ {{x \over 4}} \right]\) ,x \(\in\)
4
, where [x] denotes the
greatest integer function, then

If the function \(f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } \right.\)
is
continuous at x = 5, then the value of a – b is :-

\(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}\) equals:

Let ƒ : [–1,3] \(\to\) R be defined as
$$f(x) = \left\{ {\matrix{
{\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr
{x + \left| x \right|} & , & {1 \le x < 2} \cr
{x + \left[ x \right]} & , & {2 \le x \le 3} \...

Let ƒ : R \(\to\) R be a differentiable function
satisfying ƒ'(3) + ƒ'(2) = 0.
Then \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3)} \over {1 + f(2 - x) - f(2)}}} \right)^{{1 \over x}}}\) is equal to

\(\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}\) is :

Let S be the set of all points in (–\(\pi\), \(\pi\)) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of which of the following ?

Let f be a differentiable function such that f(1) = 2 and f '(x) = f(x) for all x \(\in\) R R. If h(x) = f(f(x)), then h'(1) is equal to :

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

If \(\alpha\) and \(\beta\) are the roots of the equation 375x2
– 25x – 2 = 0, then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\alpha ^r}} + \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{\beta ^r}} ...

\(\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}\) is :

Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x \(\in\) R. If f(x) attains maximum value at \(\alpha\) and g(x) attains
minimum value at \(\beta\), then
$$\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2}...

Let \(f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.\) and
\(g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).\)
Then, in the ...

Let [x] denote the greatest integer less than or equal to x. Then $$\mathop {\lim }\limits_{x \to 0} {{\tan \left( {\pi {{\sin }^2}x} \right) + {{\left( {\left| x \right| - \sin \left( {x\left[ x \right]} \right)} \right)}^2}} \over {{x^2}}...

Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x – \(\pi\)) cos |x| is not differentiable. Then the set K is equal to :

\(\mathop {\lim }\limits_{x \to 0} {{x\cot \left( {4x} \right)} \over {{{\sin }^2}x{{\cot }^2}\left( {2x} \right)}}\) is equal to :

Let  \(f\left( x \right) = \left\{ {\matrix{ {\max \left\{ {\left| x \right|,{x^2}} \right\}} & {\left| x \right| \le 2} \cr {8 - 2\left| x \right|} & {2 < \left| x \right| \le 4} \cr } } \right.\)
Let S be the set of points in ...

For each t \(\in\) R , let [t] be the greatest integer less than or equal to t
Then  $$\mathop {\lim }\limits_{x \to 1^ + } {{\left( {1 - \left| x \right| + \sin \left| {1 - x} \right|} \right)\sin \left( {{\pi \over 2}\left[ {1 - x} \ri...

Let f : (\(-\)1, 1) \(\to\) R be a function defined by f(x) = max \(\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}.\) If K be the set of all points at which f is not differentiable, then K has exactly -

If$$f(x) = \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^{{\raise0.5ex\hbox{$\scriptstyle 3$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scr...

Let f : R \(\to\) R be differentiable at c \(\in\) R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :

If \(\mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \over {{x^2} - {k^2}}}\), then k is :

If \(\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5\), then a + b is equal to :

\(\mathop {\lim }\limits_{x \to 0} \,\,{{{{\left( {27 + x} \right)}^{{1 \over 3}}} - 3} \over {9 - {{\left( {27 + x} \right)}^{{2 \over 3}}}}}\) equals.

If the function f defined as
\(f\left( x \right) = {1 \over x} - {{k - 1} \over {{e^{2x}} - 1}},x \ne 0,\) is continuous at
x = 0, then the ordered pair (k, f(0)) is equal to :

Let S = {(\(\lambda\), \(\mu\)) \(\in\) R \(\times\) R : f(t) = (|\(\lambda\)| e|t| \(-\) \(\mu\)). sin (2|t|), t \(\in\) R, is a differentiable function}. Then S is a subset of :

Let f(x) = \(\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {,x = 2} \cr } } \right.\)
Thevaue of k for which f s continuous at x = 2 is :

Let f(x) be a polynomial of degree \(4\) having extreme values at \(x = 1\) and \(x = 2.\)
If   \(\mathop {lim}\limits_{x \to 0} \left( {{{f\left( x \right)} \over {{x^2}}} + 1} \right) = 3\)   then f(\(-\)1) is equal to :

\(\mathop {\lim }\limits_{x \to 0} {{x\tan 2x - 2x\tan x} \over {{{\left( {1 - \cos 2x} \right)}^2}}}\) equals :

Let S = { t \(\in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)\)\(\sin \left| x \right|\) is not differentiable at t}, then the set S is equal to

For each t \(\in R\), let [t] be the greatest integer less than or equal to t.
Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \r...

The value of k for which the function
$$f\left( x \right) = \left\{ {\matrix{
{{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr
{k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} ...

\(\mathop {\lim }\limits_{x \to 3}\) \({{\sqrt {3x} - 3} \over {\sqrt {2x - 4} - \sqrt 2 }}\) is equal to :

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}\) equals

If    \(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},\) then 'a' is equal to :

If the function
f(x) = \(\left\{ {\matrix{ { - x} & {x < 1} \cr {a + {{\cos }^{ - 1}}\left( {x + b} \right),} & {1 \le x \le 2} \cr } } \right.\)
is differentiable at x = 1, then \({a \over b}\) is equal to :

Let a, b \(\in\) R, (a \(\ne\) 0). If the function f defined as
$$f\left( x \right) = \left\{ {\matrix{
{{{2{x^2}} \over a}\,\,,} & {0 \le x < 1} \cr
{a\,\,\,,} & {1 \le x < \sqrt 2 } \cr
{{{2{b^2} - 4b} \over {{x^3}}},} & ...

\(\mathop {\lim }\limits_{x \to 0} \,{{{{\left( {1 - \cos 2x} \right)}^2}} \over {2x\,\tan x\, - x\tan 2x}}\) is :

Let \(p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}\) then \(log\) \(p\) is equal to :

For \(x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,\)
and \(\,\,g\left( x \right) = f\left( {f\left( x \right)} \right),\,\,\) then :

If the function.
\(g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr } } \right.\)
is differentiable, then the value of \(k+m\) is :

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

\(\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}}\) is equal to :