Mathematics - Limits
Exam Duration: 45 Mins Total Questions : 30
The value of \(\underset{x \rightarrow 4}{lim}{|x-4|\over x-4}\)is equal to
- (a)
3
- (b)
-2
- (c)
1
- (d)
does not exist
\(\underset{x \rightarrow 0}{lim}{X\sqrt [3]{Z^2-(z-x)^2}\over[\sqrt[3]{8xz-4x^2}+\sqrt[3]{8xz}]^4}\) is equal to
- (a)
\(1\over2^{23/3}.z\)
- (b)
\(1\over 2^{23/3}.z^2\)
- (c)
\(1\over 2^{22}.z^2\)
- (d)
None of these
Let \(\alpha\) and \(\beta\) be the distinct roots of ax2+bx+c=0, then \(\underset { x\rightarrow \alpha }{ lim } \frac { 1-cos(ax^{ 2 }+bx+c) }{ (x-\alpha )^{ 2 } } \) is equal to
- (a)
\({a^2\over2}(\alpha-\beta)^2\)
- (b)
\(-{a^2\over2}(\alpha-\beta)^2\)
- (c)
\(-{a^2\over2}(\alpha+\beta)^2\)
- (d)
None of these
\(\underset { x\rightarrow 0 }{ lim } \frac { \sqrt { 1-cos2x } }{ \sqrt { 2x } } =k\) is equal to
- (a)
\(\lambda \)
- (b)
-1
- (c)
zero
- (d)
does not exist
\(\lim _{ n\rightarrow \infty }{ \left( \frac { 1-2+3-4+5-6+....-2n }{ \sqrt { ({ n }^{ 2 }+1 } +\sqrt { ({ 4n }^{ 2 }-1 } } \right) } \) is equal to
- (a)
-1
- (b)
1
- (c)
-1/3
- (d)
1/3
Let \(f(x)=\lim _{ n\rightarrow \infty }{ n({ x }^{ 1/n } } -1),x>0\) then f(xy) is equal to
- (a)
f(x)+f(y)
- (b)
f(x).f(y)
- (c)
x f(y)+y f(x)
- (d)
x f (y)
\(\lim _{ x\rightarrow 0 }{ \frac { sin(\pi { cos }^{ 2 }x) }{ { x }^{ 2 } } } \) is equal to
- (a)
\(-\pi \)
- (b)
\(\pi \)
- (c)
\(\pi /2\)
- (d)
1
\(\lim _{ n\rightarrow \infty }{ \{ \log _{ n-1 }{ (n) } \log _{ n }{ (n+1) } \log _{ n+1 }{ (n+2) } ...\log _{ n }{ k } -1({ n }^{ x })\} } \) is equal to
- (a)
n
- (b)
k
- (c)
\(\infty \)
- (d)
none of these
If \(\lim _{ x\rightarrow \infty }{ \left( 1+\frac { a }{ x } +\frac { b }{ x^{ 2 } } \right) ^{ 2x } } ={ e }^{ 2 }\) , then
- (a)
a=1,b=2
- (b)
a=2,b=1
- (c)
a=1,b∈R
- (d)
none of these
If \({ S }_{ n }=\sum _{ k=1 }^{ n }{ { a }_{ k } } \) and \(\lim _{ n\rightarrow \infty }{ { a }_{ n }=a } \) , then \(\lim _{ n\rightarrow \infty }{ \frac { { S }_{ n+1 }-{ S }_{ n } }{ \sqrt { \sum _{ k=1 }^{ n }{ k } } } } \) is equal to
- (a)
0
- (b)
\(\sqrt { 2 } a\)
- (c)
2a
If ∝ and β be the roots of ax2+bx+c=0,then \(\lim _{ x\rightarrow \infty }{ \left( 1+{ ax }^{ 2 }+bx+c \right) } ^{ 1/\left( x-\alpha \right) }\) is
- (a)
a ( ∝ -β )
- (b)
In [a ( ∝ -β ) ]
- (c)
ea ( ∝ -β )
- (d)
ea I ∝ -β I
If [x] denotes the greatest integer ≤ x, then \(\lim _{ x\rightarrow \infty }{ \frac { 1 }{ { n }^{ 3 } } \{ [{ 1 }^{ 2 }x]+[{ 2 }^{ 2 }x]+[{ 3 }^{ 2 }x] } +.....+[{ n }^{ 2 }x]\} \) equals
- (a)
x/2
- (b)
x/3
- (c)
x/6
- (d)
0
\(\lim _{ x\rightarrow 0 }{ \frac { sin[cos\quad x] }{ 1+[cos\quad x] } } \) ([.] denotes the greatest integer function)
- (a)
equal to 1
- (b)
equal to 0
- (c)
does not exist
- (d)
none of these
\(\lim _{ x\rightarrow 0 }{ \left( \frac { sin\quad x }{ x } \right) ^{ \left( \frac { sin\quad x }{ x-sin\quad x } \right) } } \) equals
- (a)
1
- (b)
e
- (c)
e-1
- (d)
e-2
\(\lim _{ x\rightarrow \infty }{ \sqrt { x } } \left( \sqrt { x+1 } -\sqrt { x } \right) \) equals
- (a)
\(\lim _{ x\rightarrow 0 }{ \frac { In(1+x)-x }{ { x }^{ 2 } } } \)
- (b)
\(\lim _{ x\rightarrow 0 }{ \frac { 1-cos\quad x }{ { x }^{ 2 } } } \)
- (c)
\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { \left( 1+x \right) } -1 }{ { x } } } \)
- (d)
\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { \left( { x }^{ 2 }+2x \right) } } } \)
If \(\lim _{ x\rightarrow 0 }{ \left( \frac { { a }^{ x }+b^{ x }+c^{ x } }{ 3 } \right) ^{ \lambda /c } } ,(a,b,c,\lambda >0)\) is equal to
- (a)
1, if λ=1
- (b)
abc, if λ=1
- (c)
abc , if λ=3
- (d)
(abc)2/3,if λ=2
If \(\lim _{ x\rightarrow { 0 }^{ + } }{ f(x) } \) =finite where \(f\left( x \right) =\frac { \sin { x+{ ae }^{ x }+{ be }^{ -x }+cIn\quad (1+x) } }{ { x }^{ 3 } } \) and a,b,c are real numbers
The value of a is
- (a)
\(-\frac { 1 }{ 2 } \)
- (b)
0
- (c)
\(\frac { 1 }{ 2 } \)
- (d)
1
If \(\lim _{ x\rightarrow { 0 }^{ + } }{ f(x) } \) =finite where \(f\left( x \right) =\frac { \sin { x+{ ae }^{ x }+{ be }^{ -x }+cIn\quad (1+x) } }{ { x }^{ 3 } } \) and a,b, c are real numbers
The value of c is
- (a)
-1/2
- (b)
1/2
- (c)
0
- (d)
2
Evaluate of the following limits.
\(\lim _{ x\rightarrow 2 }{ \left( \frac { { x }^{ 2 }-1 }{ { x }^{ 4 }-2{ x }^{ 3 }+4x-5 } \right) } \)=
- (a)
0
- (b)
5
- (c)
6
- (d)
1
Evaluate of the following limis.
\(\lim _{ x\rightarrow 0 }{ \frac { tanx }{ x } } \) is
- (a)
0
- (b)
1
- (c)
2
- (d)
Not defined
Find the derivative of the following function.
f(x)=2x2+3x-5 at x=-1
- (a)
0
- (b)
1
- (c)
-1
- (d)
-2
Find the derivative of the following function.
sinx at x=0
- (a)
1
- (b)
5
- (c)
3
- (d)
2
If y=f(x)=-cosecx.cosx,then \(\left( { \cfrac { dy }{ dx } } \right) _{ x=\cfrac { \pi }{ 2 } }\) is equal to
- (a)
0
- (b)
\(\cfrac { 1 }{ 2 } \)
- (c)
1
- (d)
3
If y=x tanx, then dy/dx is equal to
- (a)
\(\cfrac { tan\quad x }{ x-{ x }^{ 2 }-{ y }^{ 2 } } \)
- (b)
\(\cfrac { y }{ x-{ x }^{ 2 }-{ y }^{ 2 } } \)
- (c)
\(\cfrac { tan\quad x }{ y-x } \)
- (d)
\(\cfrac { cosxsinx+x }{ { cos }^{ 2 }x } \)
For f(x)=\(\cfrac { sin\quad x+cosx }{ sinx-cosx } ,\) then f\(x)=
- (a)
\(\cfrac { -2 }{ \left( sinx-cosx \right) ^{ 2 } } \)
- (b)
\(\cfrac { 2 }{ sin^{ 2 }x+sin2x } \)
- (c)
\(\cfrac { 2 }{ \left( sinx+cosx \right) ^{ 2 } } \)
- (d)
None of these
If f(x) =\(\frac { { x }_{ n }-{ a }_{ n } }{ x-a } \) for some constant 'a', then f'(a) is
- (a)
1
- (b)
0
- (c)
Does not exist
- (d)
\(\frac{1}{2}\)
\(\underset { \theta \rightarrow 0 }{ lim } \cfrac { cosecx-cotx }{ x } \) is
- (a)
\(\cfrac { -1 }{ 2 } \)
- (b)
-1
- (c)
\(\cfrac { 1 }{ 2 } \)
- (d)
-1
Let \(f\left( x \right) =\begin{cases} { x }^{ 2 }-1,\quad 0<x<2 \\ 2x+3,\quad 2\le x<3 \end{cases}\)the quadratic equation whose roots are \(\lim _{ x\rightarrow { 2 }^{ - } }{ f\left( x \right) } \) and \(\lim _{ x\rightarrow { 2 }^{ + } }{ f\left( x \right) } \) is
- (a)
x2-6x+9=0
- (b)
x2-7x+8=0
- (c)
x2-14x+49=0
- (d)
x2-10x+21=0
Statement-I: \(\lim _{ x\longrightarrow 0 }{ \frac { { e }^{ 1/x }-1 }{ { e }^{ x }+1 } } \) does not exist.
Statement-II: L.H.L. = 1and R.H.L. = -1
- (a)
Ifboth Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -I.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -I.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.