If f(x) \( = \left\{ {\begin{array}{*{20}{c}} {\rm x;if\;x\;is\;rational}\\ { \rm- x;if\;x\;is\;irrational} \end{array}} \right.\), then which of the following statement is true:

f(x) is continuous at \(x = \frac 1 2\)

\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 1 - } \frac{{\sqrt {\rm{\pi }} - \sqrt {2{\rm{si}}{{\rm{n}}^{ - 1}}{\rm{x}}} }}{{\sqrt {1 - {\rm{x}}} }}{\rm{\;}}\)is equal to

\(\sqrt {\frac{{\rm{\pi }}}{2}}\)

\(\sqrt {\frac{2}{{\rm{\pi }}}}\)

\(\frac{1}{{\sqrt {2{\rm{\pi }}} }}\)

If\(\;{\rm{f}}\left( {\rm{x}} \right) = \left[ {\rm{x}} \right] - \left[ {\frac{{\rm{x}}}{4}} \right]\) , x ∈ R, where [x] denotes the greatest integer function, then:

\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right){\rm{\;exists\;but\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 - } {\rm{f}}\left( {\rm{x}} \right)\) does not exist

Both \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to {4^ - }} {\rm{f}}\left( {\rm{x}} \right){\rm{\;and\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right){\rm{\;}}\)exist but are not equal.

\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 - } {\rm{f}}\left( {\rm{x}} \right)\;{\rm{exists\;but}}\;\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right)\) does not exist

Let S be the set of all values offor which the tangent to the curvey = f(x) = x3 – x2 – 2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (-1, f(-1)), then S is equal to:

\(\left\{ {\frac{1}{3},1} \right\}\)

\(\left\{ { - \frac{1}{3}, - 1} \right\}\)

\(\left\{ {\frac{1}{3}, - 1} \right\}\)

\(\left\{ { - \frac{1}{3},1} \right\}\)

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

Exists and equals \(\frac{1}{{2\sqrt 2 \left( {\sqrt 2 + 1} \right)}}\)

Exists and equals \(\frac{1}{{4\sqrt 2 }}\)

Exists and equals \(\frac{1}{{2\sqrt 2 \left( {\sqrt 2 + 1} \right)}}\)

Exists and equals \(\frac{1}{{2\sqrt 2 }}\)

For the function f(x) = |x – 3|, which one of the following is not correct?

The function is continuous at x = 3

The function is not continuous at x = 3

The function is continuous at x = 3

The function is differentiable at x = 0

The function is differentiable at x = -3

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

Not differentiable if f'(c) = 0

If \(f\left( x \right) = \frac{x}{2} - 1\), then on the interval [0, π] which one of the following is correct?

tan [f(x)], where [⋅] is the greatest integer function, is discontinuous but \(\frac{1}{{f\left( x \right)}}\) is continuous

tan [f(x)], where [⋅] is the greatest integer function, and \(\frac{1}{{f\left( x \right)}}\) are both continuous

tan [f(x)], where [⋅] is the greatest integer function, and f-1(x) are both continuous

tan [f(x)], where [⋅] is the greatest integer function, and \(\frac{1}{{f\left( x \right)}}\) are both discontinuous

tan [f(x)], where [⋅] is the greatest integer function, is discontinuous but \(\frac{1}{{f\left( x \right)}}\) is continuous

If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1; then the setS = {x ∈ R : f(x) = f(0)} contains exactly:

Two irrational and two rational numbers.

Two irrational and one rational number.

Let \(f\left( {x,y} \right) = \;\left\{ {\begin{array}{*{20}{c}} {\frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\;\;\;{x^2} + {y^2} \ne 0}\\ {0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x = y = 0} \end{array}} \right.\), Then

f(x, y) is continuous at the origin

f(x, y) is not differentiable at the origin

f(x, y) is continuous at the origin

Let f be a differentiable function such that \({f'}\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x}\), (x > 0) and f(1) ≠ 4. Then \(\mathop {{\rm{lim}}}\limits_{x \to {0^ + }} xf\left( {\frac{1}{x}} \right)\):

Exists and equals \(\frac{4}{{7}}\).

If f(x) = |x| + |x – 1|, then which one of the followings is correct?

f(x) is continuous at x = 0 but not at x = 1

f(x) is continuous at x = 0 and x = 1

f(x) is continuous at x = 0 but not at x = 1

f(x) is continuous at x = 1 but not at x = 0

f(x) is neither continuous at x = 0 nor at x = 1

32 + Mcqs in Limit And Continuity in Mathematics Page 1 McqOptions

1.	If f(x) \( = \left\{ {\begin{array}{*{20}{c}} {\rm x;if\;x\;is\;rational}\\ { \rm- x;if\;x\;is\;irrational} \end{array}} \right.\), then which of the following statement is true:
A.	f(x) is an odd function
B.	f(x) is continuous at \(x = \frac 1 2\)
C.	f(x) is continuous at x = 0
D.	f(x) is a periodic function
Answer» B. f(x) is continuous at \(x = \frac 1 2\)

2.	If f(4) = 4, f '(4) = 1, then \(\displaystyle\lim_{x \rightarrow 4} \dfrac{2- \sqrt{f(x)}}{2-\sqrt x}\) is equal to:
A.	-2
B.	2
C.	1
D.	-1
Answer» D. -1

3.	If the function \(f(x)=\dfrac{2x-\sin^{-1}x}{2x+\tan^{-1}x}, x\neq 0\) is continuous at each point of its domain, then the value of f(0) is:
A.	2
B.	\(\dfrac{1}{3}\)
C.	\(\dfrac{2}{3}\)
D.	\(-\dfrac{1}{3}\)
Answer» C. \(\dfrac{2}{3}\)

5.	\(\mathop {\lim }\limits_{y \to a} (\sin \frac{{y - a}}{2}\tan \frac{{\pi y}}{{2a}})\) is equal to
A.	0
B.	1
C.	\(\frac{\pi}{a}\)
D.	\(-\frac{a}{\pi}\)
Answer» E.

6.	\(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to - \infty } \left( {\frac{{\rm{n}}}{{{{\rm{n}}^2} + {1^2}}} + \frac{{\rm{n}}}{{{{\rm{n}}^2} + {2^2}}} + \frac{{\rm{n}}}{{{{\rm{n}}^2} + {3^2}}} + \ldots + \frac{1}{{5{\rm{n}}}}} \right)\) is equal to
A.	tan-1(3)
B.	tan-1(2)
C.	π/4
D.	π/2
Answer» C. π/4

7.	If\(\;{\rm{f}}\left( {\rm{x}} \right) = \left[ {\rm{x}} \right] - \left[ {\frac{{\rm{x}}}{4}} \right]\) , x ∈ R, where [x] denotes the greatest integer function, then:
A.	f is continuous at x = 4
B.	\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right){\rm{\;exists\;but\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 - } {\rm{f}}\left( {\rm{x}} \right)\) does not exist
C.	Both \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to {4^ - }} {\rm{f}}\left( {\rm{x}} \right){\rm{\;and\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right){\rm{\;}}\)exist but are not equal.
D.	\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 - } {\rm{f}}\left( {\rm{x}} \right)\;{\rm{exists\;but}}\;\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right)\) does not exist
Answer» B. \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 + } {\rm{f}}\left( {\rm{x}} \right){\rm{\;exists\;but\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 4 - } {\rm{f}}\left( {\rm{x}} \right)\) does not exist

8.	If \(f\left( x \right) = \sqrt {25 - {x^2},} \) then what is \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\) equal to?
A.	\( - \frac{1}{{\sqrt {24} }}\)
B.	\(\frac{1}{{\sqrt {24} }}\)
C.	\( - \frac{1}{{4\sqrt 3 }}\)
D.	\(\frac{1}{{4\sqrt 3 }}\)
Answer» B. \(\frac{1}{{\sqrt {24} }}\)

11.	\(\mathop {\lim }\limits_{x \to 0} \frac{{1 - {{\cos }^3}4x}}{{{x^2}}}\) is equal to
A.	0
B.	12
C.	24
D.	36
Answer» D. 36

12.	Let f(x) = 5 – \|x – 2 \| and g(x) = \|x + 1\|, x ∈ R. If f(x) attains maximum value at α and g(x) attains minimum at β, then \(\mathop {{\rm{lim}}}\limits_{x \to - \alpha \beta } \frac{{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)}}{{{x^2} - 6x + 8}}\) is equal to:
A.	1/2
B.	-3/2
C.	-1/2
D.	3/2
Answer» B. -3/2

13.	If the fourth term in the binomial expansion of \({\left( {\sqrt {\frac{1}{{{x^{1 + {\rm{lo}}{{\rm{g}}_{10}}x}}}}} + {x^{\frac{1}{{12}}}}} \right)^6}\) is equal to 200, and x > 1, then the value of x is:
A.	100
B.	10
C.	103
D.	104
Answer» C. 103

15.	If \(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 1} \frac{{{{\rm{x}}^4} - 1}}{{{\rm{x}} - 1}} = \mathop {{\rm{lim}}}\limits_{{\rm{x}} \to {\rm{k}}} \frac{{{{\rm{x}}^3} - {{\rm{k}}^3}}}{{{{\rm{x}}^2} - {{\rm{k}}^2}}}\), then k is:
A.	8/3
B.	3/8
C.	3/2
D.	4/3
Answer» B. 3/8

16.	For each t ∈ R, let [t] be the greatest integer less than or equal to t. Then, \(\mathop {{\rm{lim}}}\limits_{x \to 1 + } \frac{{\left( {1 - \left\| x \right\| + {\rm{sin}}\left\| {1 - x} \right\|} \right){\rm{sin}}\left( {\frac{\pi }{2}\left[ {1 - x} \right]} \right)}}{{\left\| {1 - x} \right\|\left[ {1 - x} \right]}}\)
A.	Equals 1
B.	Equals 0
C.	Equals -1
D.	Does not exists
Answer» C. Equals -1

19.	Examine the continuity of a function f(x) = (x - 2) (x - 3)
A.	Discontinuous at x = 2
B.	Discontinuous at x = 2, 3
C.	Continuous everywhere
D.	Discontinuous at x = 3
E.	None of these
Answer» D. Discontinuous at x = 3

20.	Let [x] denote the greatest integer less than or equal to x. Then: \(\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{{\rm{tan}}\left( {\pi {\rm{si}}{{\rm{n}}^2}x} \right) + {{(\left\| x \right\| - {\rm{sin}}\left( {x\left[ x \right]} \right))}^2}}}{{{x^2}}}\)
A.	Does not exist
B.	Equals π
C.	Equals π + 1
D.	Equals 0
Answer» B. Equals π

32.	For each x ∈ R, let [x] be the greatest integer less than or equal to x. Then \(\underset{x\to {{0}^{-}}}{\mathop{\text{lim}}}\,\frac{x\left( \left[ x \right]+\left\| x \right\| \right)\text{sin}\left[ x \right]}{\left\| x \right\|}\) is equal to:
A.	- sin 1
B.	1
C.	sin 1
D.	0
Answer» B. 1

17.	Consider the following in respect of the function \({\rm{f}}\left( {\rm{x}} \right) = {\rm{\;}}\left\{ {\begin{array}{*{20}{c}} {2 + {\rm{x}},{\rm{\;\;x}} \ge 0}\\ {2 - {\rm{x}},{\rm{\;\;x}} < 0} \end{array}} \right.\)1. \(\mathop {\lim }\limits_{{\rm{x}} \to 1} {\rm{f}}\left( {\rm{x}} \right)\) does not exist.2. f(x) is differentiable at x = 03. f(x) is continuous at x = 0Which of the above statement is/are correct?
A.	1 only
B.	3 only
C.	2 and 3 only
D.	1 and 3 only
Answer» C. 2 and 3 only

18.	Let K be the set of all real values of x where the function f(x) = sin\|x\|-\|x\| + 2(x - π) cos\|x\| is not differentiable. Then the set K is equal to:
A.	ϕ (an empty set)
B.	{π}
C.	{0}
D.	{0, π}
Answer» B. {π}

21.	Let f : R → R be differentiable at c ∈ R and f(c) = 0. If g(x) = \|f(x)\|, then at x = c, g is:
A.	Not differentiable if f'(c) = 0
B.	Differentiable if f'(c) ≠ 0
C.	Differentiable if f'(c) = 0
D.	Not differentiable
Answer» D. Not differentiable

23.	If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:
A.	x2 + y2 – 4x2y2 = 0
B.	x2 + y2 – 2xy = 0
C.	x2 + y2 – 16x2y2 = 0
D.	x2 + y2 – 2x2y2 = 0
Answer» B. x2 + y2 – 2xy = 0

24.	If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1; then the setS = {x ∈ R : f(x) = f(0)} contains exactly:
A.	Four irrational numbers.
B.	Four rational numbers.
C.	Two irrational and two rational numbers.
D.	Two irrational and one rational number.
Answer» E.

26.	For what value of k is the function \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} {2x + \frac{1}{4},\;\;\;\;\;x < 0}\\ {k,\;\;\;\;\;\;\;x = 0}\\ {{{\left( {x + \frac{1}{2}} \right)}^2},\;\;\;\;\;\;\;x > 0} \end{array}} \right.\) continuous ?
A.	\(\frac{1}{4}\)
B.	\(\frac{1}{2}\)
C.	1
D.	2
Answer» B. \(\frac{1}{2}\)

27.	\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 0} \frac{{{\rm{si}}{{\rm{n}}^2}{\rm{x}}}}{{\sqrt 2 - \sqrt {1 + {\rm{cos\;x}}} }}\) equals:
A.	\(4\sqrt 2\)
B.	\(\sqrt 2\)
C.	\(2\sqrt 2\)
D.	4
Answer» B. \(\sqrt 2\)

28.	Let g be the greatest integer function. Then the function f(x) = (g(x))2 - g(x) is discontinuous at
A.	all integers
B.	all integers except 0 and 1
C.	all integers except 0
D.	all integers except 1
Answer» E.

29.	Let f be a differentiable function such that \({f'}\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x}\), (x > 0) and f(1) ≠ 4. Then \(\mathop {{\rm{lim}}}\limits_{x \to {0^ + }} xf\left( {\frac{1}{x}} \right)\):
A.	Exists and equals \(\frac{4}{{7}}\).
B.	Exists and equals 4.
C.	Does not exist.
D.	Exists and equals 0.
Answer» C. Does not exist.

30.	Consider the function\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{sin2x}}{{5x}}\;\;\;\;\;\;\;\;\;if\;x \ne 0\;\;\;\;\;\;\;\;\;\;}\\ {\frac{2}{{15}}\;\;\;\;\;if\;x = 0} \end{array}} \right.\)Which one of the following is correct in respect of the function?
A.	It is not continuous at x = 0
B.	it is continuous at every x
C.	It is not continuous at x = π
D.	It is continuous at x = 0
Answer» B. it is continuous at every x

Explore topic-wise MCQs in Mathematics.

If f(x) \( = \left\{ {\begin{array}{*{20}{c}} {\rm x;if\;x\;is\;rational}\\ { \rm- x;if\;x\;is\;irrational} \end{array}} \right.\), then which of the following statement is true:

If f(4) = 4, f '(4) = 1, then \(\displaystyle\lim_{x \rightarrow 4} \dfrac{2- \sqrt{f(x)}}{2-\sqrt x}\) is equal to:

If the function \(f(x)=\dfrac{2x-\sin^{-1}x}{2x+\tan^{-1}x}, x\neq 0\) is continuous at each point of its domain, then the value of f(0) is:

\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 1 - } \frac{{\sqrt {\rm{\pi }} - \sqrt {2{\rm{si}}{{\rm{n}}^{ - 1}}{\rm{x}}} }}{{\sqrt {1 - {\rm{x}}} }}{\rm{\;}}\)is equal to

\(\mathop {\lim }\limits_{y \to a} (\sin \frac{{y - a}}{2}\tan \frac{{\pi y}}{{2a}})\) is equal to

\(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to - \infty } \left( {\frac{{\rm{n}}}{{{{\rm{n}}^2} + {1^2}}} + \frac{{\rm{n}}}{{{{\rm{n}}^2} + {2^2}}} + \frac{{\rm{n}}}{{{{\rm{n}}^2} + {3^2}}} + \ldots + \frac{1}{{5{\rm{n}}}}} \right)\) is equal to

If\(\;{\rm{f}}\left( {\rm{x}} \right) = \left[ {\rm{x}} \right] - \left[ {\frac{{\rm{x}}}{4}} \right]\) , x ∈ R, where [x] denotes the greatest integer function, then:

If \(f\left( x \right) = \sqrt {25 - {x^2},} \) then what is \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right) - f\left( 1 \right)}}{{x - 1}}\) equal to?

Let S be the set of all values offor which the tangent to the curvey = f(x) = x3 – x2 – 2x at (x, y) is parallel to the line segment joining the points (1, f(1)) and (-1, f(-1)), then S is equal to:

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

\(\mathop {\lim }\limits_{x \to 0} \frac{{1 - {{\cos }^3}4x}}{{{x^2}}}\) is equal to

Let f(x) = 5 – |x – 2 | and g(x) = |x + 1|, x ∈ R. If f(x) attains maximum value at α and g(x) attains minimum at β, then \(\mathop {{\rm{lim}}}\limits_{x \to - \alpha \beta } \frac{{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)}}{{{x^2} - 6x + 8}}\) is equal to:

If the fourth term in the binomial expansion of \({\left( {\sqrt {\frac{1}{{{x^{1 + {\rm{lo}}{{\rm{g}}_{10}}x}}}}} + {x^{\frac{1}{{12}}}}} \right)^6}\) is equal to 200, and x > 1, then the value of x is:

For the function f(x) = |x – 3|, which one of the following is not correct?

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

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

Examine the continuity of a function f(x) = (x - 2) (x - 3)

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

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

If \(f\left( x \right) = \frac{x}{2} - 1\), then on the interval [0, π] which one of the following is correct?

If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is:

If f(x) is a non-zero polynomial of degree four, having local extreme points at x = -1, 0, 1; then the setS = {x ∈ R : f(x) = f(0)} contains exactly:

Let \(f\left( {x,y} \right) = \;\left\{ {\begin{array}{*{20}{c}} {\frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\;\;\;{x^2} + {y^2} \ne 0}\\ {0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x = y = 0} \end{array}} \right.\), Then

For what value of k is the function \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} {2x + \frac{1}{4},\;\;\;\;\;x < 0}\\ {k,\;\;\;\;\;\;\;x = 0}\\ {{{\left( {x + \frac{1}{2}} \right)}^2},\;\;\;\;\;\;\;x > 0} \end{array}} \right.\) continuous ?

\(\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 0} \frac{{{\rm{si}}{{\rm{n}}^2}{\rm{x}}}}{{\sqrt 2 - \sqrt {1 + {\rm{cos\;x}}} }}\) equals:

Let g be the greatest integer function. Then the function f(x) = (g(x))2 - g(x) is discontinuous at

Let f be a differentiable function such that \({f'}\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x}\), (x > 0) and f(1) ≠ 4. Then \(\mathop {{\rm{lim}}}\limits_{x \to {0^ + }} xf\left( {\frac{1}{x}} \right)\):

Consider the function\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{sin2x}}{{5x}}\;\;\;\;\;\;\;\;\;if\;x \ne 0\;\;\;\;\;\;\;\;\;\;}\\ {\frac{2}{{15}}\;\;\;\;\;if\;x = 0} \end{array}} \right.\)Which one of the following is correct in respect of the function?

If f(x) = |x| + |x – 1|, then which one of the followings is correct?

For each x ∈ R, let [x] be the greatest integer less than or equal to x. Then \(\underset{x\to {{0}^{-}}}{\mathop{\text{lim}}}\,\frac{x\left( \left[ x \right]+\left| x \right| \right)\text{sin}\left[ x \right]}{\left| x \right|}\) is equal to: