Mathematics - Limits
Exam Duration: 45 Mins Total Questions : 30
\(\underset { X\ \rightarrow \ 1 }{ lim } \)[x - 1], where [.] is greatest integer function, is
- (a)
1
- (b)
2
- (c)
0
- (d)
does not exist
\(\underset{a \rightarrow \infty}{lim}[{1\over 1-a}+{8\over 1-a^4}+...+{a^3\over1-a^4}]\)is equal to
- (a)
\(1\over3\)
- (b)
\(1\over4\)
- (c)
\(-{1\over3}\)
- (d)
\(-{1\over4}\)
\(\underset{x \rightarrow {\pi\over2}}{lim}[1^{1/cos^2\ x}+2^{1\cos^2\ x}+...+n^{1/cos^2\ x}]^{cos^2\ x}\) is equal to
- (a)
o
- (b)
n
- (c)
\(\infty\)
- (d)
\(n(n+1)\over2\)
\(\lim _{ n\rightarrow \infty }{ \sum _{ x=1 }^{ 20 }{ { cos }^{ 2n } } } (x-10)\) is equal to
- (a)
0
- (b)
1
- (c)
19
- (d)
20
The value of \(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1/2(1-cos2x) } }{ x } } \) is equal to
- (a)
1
- (b)
-1
- (c)
0
- (d)
none of these
The value of \(\lim _{ x\rightarrow \infty }{ \left( \frac { { x }^{ 2 }sin(1/x)-x }{ 1-\left| x \right| } \right) } \) is
- (a)
0
- (b)
1
- (c)
-1
- (d)
none of these
\(\lim _{ x\rightarrow 0 }{ \frac { x\sqrt { { y }^{ 2 }-(y-x)^{ 2 } } }{ \left( \left( \sqrt { (8xy-{ 4x }^{ 2 } } \right) +\sqrt { (8xy)^{ 3 } } \right) } } \) is equal to
- (a)
1/4
- (b)
1/2
- (c)
\(1/2\sqrt { 2 } \)
- (d)
none of these
If \(\lim _{ x\rightarrow 0 }{ \frac { \left( (a-n)nx-tan\quad x \right) sin\quad nx }{ { x }^{ 2 } } } =0\) where n is non zero real number,then a is equal to
- (a)
0
- (b)
\(\frac { n+1 }{ n } \)
- (c)
n
- (d)
\(n+\frac { 1 }{ n } \)
The value of \(\lim _{ x\rightarrow 0 }{ \left[ { x }^{ 2 }+x+sin\quad x \right] } \) is (where [.] denotes the greatest integer function)
- (a)
does not exist
- (b)
is equal to zero
- (c)
-1
- (d)
none of these
The graph of the function y = f(x) has a unique tangent at the point (a, 0) through which the graph passes, Then \(\lim _{ x\rightarrow a }{ \frac { \log _{ e }{ \{ 1+6f(x)\} } }{ 3f(x) } } \) is
- (a)
0
- (b)
1
- (c)
2
- (d)
none of these
\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ n }sin^{ n }\quad x }{ { x }^{ n }-sin^{ n }\quad x } } \) is non finite,then n must be equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
none of these
\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ n }-{ sin\quad x }^{ n } }{ { x }-{ sin^{ n }\quad x } } } \) Is non-zero finite, then n must be equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
none of these
Let \(f(x)=\lim _{ n\rightarrow \infty }{ \frac { { x }^{ 2n }-1 }{ { x }^{ 2n }+1 } } \) then
- (a)
f(x)=1 for |x|>1
- (b)
f(x)=-1 for |x|<1
- (c)
f(x) is not defined for any value of x
- (d)
f(x)=1 for |x|=1
If,\(\lim _{ x\rightarrow 0 }{ (cos\quad x+\alpha \quad sin\quad bx)^{ 1/x } } ={ e }^{ 2 }\) then the values of ∝ and b
- (a)
a=1, b=2
- (b)
a=2,b=1/2
- (c)
\(a=2\sqrt { 2 } ,b=\frac { 1 }{ \sqrt { 2 } } \)
- (d)
a=4 ,b=2
Evaluate of the following limits.
\(\lim _{ x\rightarrow 1 }{ \left[ \frac { x^{ 2 }+1 }{ x+100 } \right] } \)=
- (a)
1
- (b)
\(\frac{2}{101}\)
- (c)
0
- (d)
\(\frac{101}{2}\)
Evaluate of the following limits.
\(\lim _{ x\rightarrow 2 }{ \left( \frac { { x }^{ 2 }-3x+2 }{ { x }^{ 2 }+x-6 } \right) }\)=
- (a)
\(\frac{1}{5}\)
- (b)
\(\frac{2}{5}\)
- (c)
\(\frac{3}{5}\)
- (d)
1
L=\(\lim _{ x\rightarrow -\infty }{ \sqrt { [{ x }^{ 2 }-x+1 } } \)-(ax+b)]=0, then the values of a and b respectively are
- (a)
1,-\(\frac{1}{2}\)
- (b)
1,1
- (c)
\(\frac{1}{2}\),1
- (d)
-\(\frac{1}{2}\),1
\(\lim _{ x\rightarrow 1 }{ \left[ \left( \frac { 4 }{ { x }^{ 2 }-x^{ -1 } } -\frac { 1-3x+{ x }^{ 2 } }{ 1-x^{ 3 } } \right) ^{ -1 }+3\left( \frac { { x }^{ 4 }-1 }{ { x }^{ 3 }-{ x }^{ -1 } } \right) \right] } \)is
- (a)
2
- (b)
3
- (c)
4
- (d)
1
Evaluate of the following limis.
\(\lim _{ x\rightarrow 0 }{ \frac { \sqrt { 1-\sqrt { cos2x } } }{ x } } \) is
- (a)
0
- (b)
1
- (c)
-1
- (d)
does not exist
\(\lim _{ x\rightarrow \frac { \pi }{ 4 } }{ \frac { cosx-sinx }{ \left( \frac { \pi }{ 4 } -x \right) (cosx+sinx) } } \) is
- (a)
0
- (b)
1
- (c)
-1
- (d)
2
\(\lim _{ x\rightarrow 0 }{ \frac { sin3x }{ sin2x } } \)=
- (a)
1
- (b)
\(\frac{1}{2}\)
- (c)
\(\frac{3}{2}\)
- (d)
0
Find the derivative of the following function.
sinx at x=0
- (a)
1
- (b)
5
- (c)
3
- (d)
2
Compute the dervivative of 6x100-x55+x
- (a)
x99-x54+1
- (b)
100x99-55x4+1
- (c)
600x99-55x54+1
- (d)
None of these
\(\lim _{ x\rightarrow 0 }{ \frac { tan2x-x }{ 3x-sinx } } \) is
- (a)
2
- (b)
\(1\over2\)
- (c)
\(-1\over2\)
- (d)
\(1\over4\)
Statement-I: The derivative of h(x)=\(\frac { x+cosx }{ tanx } \) is \(\frac { (1-sinx)tanx-(x+cosx)sec^{ 2 }x }{ (tanx)^{ 2 } } \)
Statement-II: We use the rule \(\left( \frac { u }{ v } \right) '=\frac { u'v-uv' }{ (v)^{ 2 } } \)
- (a)
If both 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
Ifj(x) = 1+x\(\frac { { x }^{ 2 } }{ 2 } +...\frac { { x }^{ 100 } }{ 100 } \) thenf'(1) is
- (a)
\(\frac{1}{100}\)
- (b)
100
- (c)
Does not exist
- (d)
0
If y =\(\frac { sinx+cosx }{ sinx-cosx } ,\) then \(\frac{dy}{dx}\)at x=0 is
- (a)
-2
- (b)
0
- (c)
\(\frac{1}{2}\)
- (d)
Does not exist