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Limits, Continuity and Differentiability

328 Questions
All Exams JEE Mains JEE Advanced
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Evening Shift
If $f:R \to R$ is given by $f(x) = x + 1$, then the value of $\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right]$ is :
A.
${3 \over 2}$
B.
${5 \over 2}$
C.
${1 \over 2}$
D.
${7 \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 20th July Morning Shift
Let a function f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right.$

where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:
A.
4
B.
3
C.
2
D.
5
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let f : R $ \to $ R be a function defined as

$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\, = 0 \hfill \cr {{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill \cr} \right.$

If f is continuous at x = 0, then the value of a + b is equal to :
A.
$-$3
B.
$-$2
C.
$ - {5 \over 2}$
D.
$ - {3 \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
If $\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}}$ is equal to L, then the value of (6L + 1) is
A.
${1 \over 6}$
B.
${1 \over 2}$
C.
6
D.
2
2021 JEE Mains MCQ
JEE Main 2021 (Online) 18th March Morning Shift
If $f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right.$ is differentiable at every point of the domain, then the values of a and b are respectively :
A.
${1 \over 2},{1 \over 2}$
B.
${1 \over 2}, - {3 \over 2}$
C.
${5 \over 2}, - {3 \over 2}$
D.
$ - {1 \over 2},{3 \over 2}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
The value of the limit

$\mathop {\lim }\limits_{\theta \to 0} {{\tan (\pi {{\cos }^2}\theta )} \over {\sin (2\pi {{\sin }^2}\theta )}}$ is equal to :
A.
0
B.
$-$${1 \over 2}$
C.
${1 \over 4}$
D.
$-$${1 \over 4}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Evening Shift
The value of $\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :
A.
r
B.
${r \over 2}$
C.
0
D.
2r
2021 JEE Mains MCQ
JEE Main 2021 (Online) 17th March Morning Shift
The value of
$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$, where [ x ] denotes the greatest integer $ \le $ x is :
A.
$\pi$
B.
${\pi \over 4}$
C.
${\pi \over 2}$
D.
0
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
Let f : S $ \to $ S where S = (0, $\infty $) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $ \to $ R be defined as g(x) = loge f(x), then the value of |g''(5) $-$ g''(1)| is equal to :
A.
1
B.
${{187} \over {144}}$
C.
${{197} \over {144}}$
D.
${{205} \over {144}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Evening Shift
Let $\alpha$ $\in$ R be such that the function $f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$ is continuous at x = 0, where {x} = x $-$ [ x ] is the greatest integer less than or equal to x. Then :
A.
no such $\alpha$ exists
B.
$\alpha$ = 0
C.
$\alpha$ = ${\pi \over 4}$
D.
$\alpha$ = ${\pi \over {\sqrt 2 }}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
Let ${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $. Then $\mathop {\lim }\limits_{k \to \infty } {S_k}$ is equal to :
A.
${\cot ^{ - 1}}\left( {{3 \over 2}} \right)$
B.
${\pi \over 2}$
C.
tan$-$1 (3)
D.
${\tan ^{ - 1}}\left( {{3 \over 2}} \right)$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 16th March Morning Shift
Let the functions f : R $ \to $ R and g : R $ \to $ R be defined as :

$f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.$ and

$g(x) = \left\{ {\matrix{ {{x^3},} & {x < 1} \cr {3x - 2,} & {x \ge 1} \cr } } \right.$

Then, the number of points in R where (fog) (x) is NOT differentiable is equal to :
A.
0
B.
3
C.
1
D.
2
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4.

Then $\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$ equals :
A.
4 $-$ 2a
B.
2a + 4
C.
a + 4
D.
2a $-$ 4
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let $f(x) = {\sin ^{ - 1}}x$ and $g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$. If $g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$, then the domain of the function fog is :
A.
$( - \infty , - 2] \cup \left[ { - {4 \over 3},\infty } \right)$
B.
$( - \infty , - 2] \cup [ - 1,\infty )$
C.
$( - \infty , - 2] \cup \left[ { - {3 \over 2},\infty } \right)$
D.
$( - \infty , - 1] \cup [2,\infty )$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let f : R $ \to $ R be defined as

$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$ If f(x) is continuous on R, then a + b equals :
A.
$-$3
B.
3
C.
$-$1
D.
1
2021 JEE Mains MCQ
JEE Main 2021 (Online) 26th February Morning Shift
The value of $\mathop {\lim }\limits_{h \to 0} 2\left\{ {{{\sqrt 3 \sin \left( {{\pi \over 6} + h} \right) - \cos \left( {{\pi \over 6} + h} \right)} \over {\sqrt 3 h\left( {\sqrt 3 \cosh - \sinh } \right)}}} \right\}$ is :
A.
${4 \over 3}$
B.
${2 \over 3}$
C.
${3 \over 4}$
D.
${2 \over {\sqrt 3 }}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 25th February Morning Shift
$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$ is equal to :
A.
${{1 \over 2}}$
B.
1
C.
0
D.
${{1 \over e}}$
2021 JEE Mains MCQ
JEE Main 2021 (Online) 24th February Morning Shift
If f : R $ \to $ R is a function defined by f(x)= [x - 1] $\cos \left( {{{2x - 1} \over 2}} \right)\pi $, where [.] denotes the greatest integer function, then f is :
A.
continuous for every real x
B.
discontinuous at all integral values of x except at x = 1
C.
discontinuous only at x = 1
D.
continuous only at x = 1
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Evening Slot
Let f : R $ \to $ R be a function defined by
f(x) = max {x, x2}. Let S denote the set of all points in R, where f is not differentiable. Then :
A.
{0, 1}
B.
{0}
C.
$\phi $(an empty set)
D.
{1}
2020 JEE Mains MCQ
JEE Main 2020 (Online) 6th September Evening Slot
For all twice differentiable functions f : R $ \to $ R,
with f(0) = f(1) = f'(0) = 0
A.
f''(x) $ \ne $ 0, at every point x $ \in $ (0, 1)
B.
f''(x) = 0, for some x $ \in $ (0, 1)
C.
f''(0) = 0
D.
f''(x) = 0, at every point x $ \in $ (0, 1)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Evening Slot
$\mathop {\lim }\limits_{x \to 0} {{x\left( {{e^{\left( {\sqrt {1 + {x^2} + {x^4}} - 1} \right)/x}} - 1} \right)} \over {\sqrt {1 + {x^2} + {x^4}} - 1}}$
A.
is equal to 0.
B.
is equal to $\sqrt e $.
C.
is equal to 1.
D.
does not exist.
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If the function
$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr {{k_2}\cos x,} & {x > \pi } \cr } } \right.$ is
twice differentiable, then the ordered pair (k1, k2) is equal to :
A.
$\left( {{1 \over 2},-1} \right)$
B.
(1, 1)
C.
(1, 0)
D.
$\left( {{1 \over 2},1} \right)$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 5th September Morning Slot
If $\alpha $ is positive root of the equation, p(x) = x2 - x - 2 = 0, then

$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}$ is equal to :
A.
${1 \over \sqrt2}$
B.
${1 \over 2}$
C.
${3 \over \sqrt2}$
D.
${3 \over 2}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
Let $f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)$ be a differentiable function such that f(1) = e and
$\mathop {\lim }\limits_{t \to x} {{{t^2}{f^2}(x) - {x^2}{f^2}(t)} \over {t - x}} = 0$. If f(x) = 1, then x is equal to :
A.
${1 \over e}$
B.
e
C.
${1 \over 2e}$
D.
2e
2020 JEE Mains MCQ
JEE Main 2020 (Online) 4th September Evening Slot
The function
$f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} \cr {{1 \over 2}\left( {\left| x \right| - 1} \right),} & {\left| x \right| > 1} \cr } } \right.$ is :
A.
continuous on R–{–1} and differentiable on R–{–1, 1}
B.
both continuous and differentiable on R–{1}
C.
both continuous and differentiable on R–{–1}
D.
continuous on R–{1} and differentiable on R–{–1, 1}
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Evening Slot
$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}$ ($a$ $ \ne $ 0) is equal to :
A.
$\left( {{2 \over 9}} \right){\left( {{2 \over 3}} \right)^{{1 \over 3}}}$
B.
$\left( {{2 \over 3}} \right){\left( {{2 \over 9}} \right)^{{1 \over 3}}}$
C.
${\left( {{2 \over 3}} \right)^{{4 \over 3}}}$
D.
${\left( {{2 \over 9}} \right)^{{4 \over 3}}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 3rd September Morning Slot
Let [t] denote the greatest integer $ \le $ t. If for some
$\lambda $ $ \in $ R - {1, 0}, $\mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda - x + \left[ x \right]}}} \right|$ = L, then L is equal to :
A.
1
B.
2
C.
0
D.
${1 \over 2}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$ is equal to :
A.
2
B.
1
C.
$e$
D.
$e$2
2020 JEE Mains MCQ
JEE Main 2020 (Online) 2nd September Morning Slot
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 $a$, b, c $ \in $ R and f'(0) + f'(2) = e, then the value of of $a$ is :
A.
${e \over {{e^2} - 3e - 13}}$
B.
${1 \over {{e^2} - 3e + 13}}$
C.
${e \over {{e^2} - 3e + 13}}$
D.
${e \over {{e^2} + 3e + 13}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Evening Slot
Let [t] denote the greatest integer $ \le $ t and $\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A$.
Then the function, f(x) = [x2]sin($\pi $x) is discontinuous, when x is equal to :
A.
$\sqrt {A + 1} $
B.
$\sqrt {A + 5} $
C.
$\sqrt {A + 21} $
D.
$\sqrt {A} $
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
If $f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x < 0} \cr {b\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;} & {x = 0} \cr {{{{{\left( {x + 3{x^2}} \right)}^{{1 \over 3}}} - {x^{ {1 \over 3}}}} \over {{x^{{4 \over 3}}}}};} & {x > 0} \cr } } \right.$
is continuous at x = 0, then a + 2b is equal to :
A.
0
B.
-1
C.
-2
D.
1
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $ \in $ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $ \in $ (a, b), ${{f(c) - f(a)} \over {f(b) - f(c)}}$ is greater than :
A.
1
B.
${{b - c} \over {c - a}}$
C.
${{b + a} \over {b - a}}$
D.
${{c - a} \over {b - c}}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Evening Slot
Let S be the set of all functions ƒ : [0,1] $ \to $ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $ \in $ (0,1), depending on ƒ, such that
A.
$\left| {f(c) - f(1)} \right| < \left| {f'(c)} \right|$
B.
$\left| {f(c) + f(1)} \right| < \left( {1 + c} \right)\left| {f'(c)} \right|$
C.
$\left| {f(c) - f(1)} \right| < \left( {1 - c} \right)\left| {f'(c)} \right|$
D.
None
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Morning Slot
$\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}}$ is equal to
A.
e
B.
e2
C.
${1 \over {{e^2}}}$
D.
${1 \over e}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
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 $\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)} \over {{x^2} - 6x + 8}}$ is equal to :
A.
${1 \over 2}$
B.
$-{1 \over 2}$
C.
${3 \over 2}$
D.
$-{3 \over 2}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
$\mathop {\lim }\limits_{x \to 0} {{x + 2\sin x} \over {\sqrt {{x^2} + 2\sin x + 1} - \sqrt {{{\sin }^2}x - x + 1} }}$ is :
A.
6
B.
1
C.
3
D.
2
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
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}} $ is equal to :
A.
${7 \over {116}}$
B.
${{29} \over {348}}$
C.
${1 \over {12}}$
D.
${{21} \over {346}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
If $\mathop {\lim }\limits_{x \to 1} {{{x^2} - ax + b} \over {x - 1}} = 5$, then a + b is equal to :
A.
1
B.
- 4
C.
- 7
D.
5
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
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$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}}}} & {,x > 0} \cr } } \right.$
is continuous at x = 0, then the ordered pair (p, q) is equal to
A.
$\left( { - {3 \over 2}, - {1 \over 2}} \right)$
B.
$\left( { - {1 \over 2},{3 \over 2}} \right)$
C.
$\left( { - {3 \over 2}, {1 \over 2}} \right)$
D.
$\left( { {5 \over 2}, {1 \over 2}} \right)$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
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 :
A.
differentiable if f '(c) = 0
B.
differentiable if f '(c) $ \ne $ 0
C.
not differentiable
D.
not differentiable if f '(c) = 0
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
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 :
A.
${3 \over 2}$
B.
${8 \over 3}$
C.
${4 \over 3}$
D.
${3 \over 8}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
If $f(x) = [x] - \left[ {{x \over 4}} \right]$ ,x $ \in $ 4 , where [x] denotes the greatest integer function, then
A.
Both $\mathop {\lim }\limits_{x \to 4 - } f(x)$ and $\mathop {\lim }\limits_{x \to 4 + } f(x)$ exist but are not equal
B.
f is continuous at x = 4
C.
$\mathop {\lim }\limits_{x \to 4 + } f(x)$ exists but $\mathop {\lim }\limits_{x \to 4 - } f(x)$ does not exist
D.
$\mathop {\lim }\limits_{x \to 4 - } f(x)$ exists but $\mathop {\lim }\limits_{x \to 4 + } f(x)$ does not exist
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
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 :-
A.
${2 \over {\pi - 5 }}$
B.
${2 \over {5 - \pi }}$
C.
${-2 \over {\pi + 5 }}$
D.
${2 \over {\pi + 5 }}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
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 :
A.
{10,15}
B.
{5,10,15,20}
C.
{10}
D.
{5,10,15}
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
If the function ƒ defined on , $\left( {{\pi \over 6},{\pi \over 3}} \right)$ by $$f(x) = \left\{ {\matrix{ {{{\sqrt 2 {\mathop{\rm cosx}\nolimits} - 1} \over {\cot x - 1}},} & {x \ne {\pi \over 4}} \cr {k,} & {x = {\pi \over 4}} \cr } } \right.$$ is continuous, then k is equal to
A.
1
B.
1 / $\sqrt 2$
C.
${1 \over 2}$
D.
2
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
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
A.
e
B.
e2
C.
e–1
D.
1
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
Let ƒ : [–1,3] $ \to $ R be defined as

$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr {x + \left| x \right|} & , & {1 \le x < 2} \cr {x + \left[ x \right]} & , & {2 \le x \le 3} \cr } } \right.$

where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:
A.
only three points
B.
four or more points
C.
only two points
D.
only one point
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
$\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}$ equals:
A.
$ \sqrt 2$
B.
$2 \sqrt 2$
C.
4
D.
$4 \sqrt 2$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
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 :
A.
4e
B.
2e2
C.
4e2
D.
2e
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
$\mathop {\lim }\limits_{x \to {1^ - }} {{\sqrt \pi - \sqrt {2{{\sin }^{ - 1}}x} } \over {\sqrt {1 - x} }}$ is equal to :
A.
$\sqrt {{2 \over \pi }} $
B.
${1 \over {\sqrt {2\pi } }}$
C.
$\sqrt {{\pi \over 2}} $
D.
$\sqrt \pi $