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11th Maths Weekly Test -1( Differential Calculus - Limits and Continuity )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -1( Differential Calculus - Limits and Continuity )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

A function f(x) is said to be continuous at x=a if \(\lim _{ x\rightarrow a }{ f\left( x \right) } \)is equal to

\(f\left( a \right) \)
\(f\left( -a \right) \)
\(2f\left( a \right) \)
\(f\left( \frac { 1 }{ a } \right) \)

Find the odd one of the following

x²
x⁴
cos x
sin x

If f : \(R \rightarrow R\) is defined by f(x)=\(\left\lfloor x-3 \right\rfloor +|x-4|\) for \(x \in R\), then \(\underset{x\rightarrow 3^-}{lim}f(x)\) is equal to

-2
-1
0
1

Find the odd one of out of the following

tan x
\(\frac{1}{x}\)
\(\\ \\ \\ \\ \\ \\ \\ \frac { { x }^{ 2 }+5x+4 }{ { x }^{ 2 }+4x+4 } \)
cos x

Find the odd one out of the following

sin x=2
e^x=-2
tan x=7
|x|=1

Choose the incorrect pair:

1. \(\underset { x\rightarrow 0 }{ lim } (1+x)^{\frac{1}{x}}\) e

2. \(\underset { x\rightarrow 0 }{ lim }\ cos\ x\) 1

3. \(\underset { x\rightarrow 0 }{ lim } \frac { { a }^{ x }-1 }{ x } \) log a

4. \(\\ \underset { x\rightarrow 0 }{ lim } \frac { { (1+x) }^{ n }-1 }{ n } \) n

\(\underset { x\rightarrow 0 }{ lim } ({ 1+x) }^{ \frac { 1 }{ x } }\ -\ e\)
\(\underset { x\rightarrow 0 }{ lim } cos\ x\ -\ 1\)
\(\underset { x\rightarrow 0 }{ lim } \frac { { a }^{ x }-1 }{ x } \ -\ log\ a\)
\(\\ \underset { x\rightarrow 0 }{ lim } \frac { { (1+x) }^{ n }-1 }{ n } \ -\ n\)

\(\underset{x \rightarrow \infty}{lim}{\sqrt{x^2-1}\over 2x-1}=\)

1
0
-1
\(1\over 2\)

\(\underset{x \rightarrow \infty}{lim}{a^x-b^x\over x}=\)

log ab
log\(({a\over b})\)
log\(({b\over a})\)
\({a\over b}\)

For what values of x is the rate of increase of x³ - 2x² + 3x + 8 is twice the rate of increase of x?

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

\(\underset{x\rightarrow {\pi/2}}{lim}{2x-\pi\over cosx} \)

2
1
-2
0

Part - B

Generic 2 - Answer All Question

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ \frac { 2 }{ 3 } }-9 }{ x-27 } } \)
Evaluate the following limits :\(\underset{\alpha\rightarrow 0}{lim}{{sin (\alpha^n)}\over( sin \alpha )^m}\)

Compute\(lim_{x\rightarrow-2}(-{3\over 2}x)\)

For what value of k is the function \(f\left( x \right) =\begin{cases} \frac { \sin { 5x } }{ 3x } \quad if\quad x\neq 0 \\ k,\quad \quad \quad if\quad x=0 \end{cases}\) is continuous at x = 0.

Find \(\underset { x\rightarrow 1 }{ lim } \) fix), if \(f(x)=\{ \begin{matrix} { x }^{ 2 }-1 & x\le 1 \\ -{ x }^{ 2 }-1 & x>1 \end{matrix}\)

Part - C

Generic 3 - Answer All Question

Evaluate the following limits :
\(\underset{x\rightarrow 1}{lim}{\sqrt{x}-x^2\over 1-\sqrt{x}}\)
\(It\quad \lim _{ x\rightarrow a }{ \frac { { x }^{ 9 }-{ a }^{ 9 } }{ x-a } } =9,\)find all possible values of a.

Calculate \(lim_{x \rightarrow \infty}{1-x^3\over 3x+2}\)

Let f(x)=\(\begin{cases} 0, & if\quad x<0 \\ { x }^{ 2 }, & if\quad 0\le x\le 2 \\ 4, & if\quad x\ge 2 \end{cases}\).Graph the function. Show that f(x) continuous on\((-\infty,\infty)\).

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin|x|\over x\)

Part - D

Generic 5 - Answer All Question

Evaluate \(\lim _{ x\rightarrow \sqrt { 2 } }{ \frac { { x }^{ 9 }-3{ x }^{ 8 }+{ x }^{ 6 }-9{ x }^{ 4 }-4{ x }^{ 2 }-16x+84 }{ { x }^{ 5 }-3{ x }^{ 4 }-4x+12 } } \)

A function f is defined as follows :
f(x)=\(\begin{cases} 0, & for\ x<0; \\ x, & for\ 0\le x<1; \\ -{ x }^{ 2 }+4x-2, & for\ 1\le x<3; \\ 4-x, & for\ x\ge 3 \end{cases}\)
Is the function continuous?

Evaluate the following limits :
\(\underset{x\rightarrow0}{lim}{\sqrt{1-x}-1\over x^2}\)

Evaluate the following limits :\(\underset{x \rightarrow \infty}{lim}\left({x^2-2x+1\over x^2-4x+2}\right)^x\)

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