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

St. Britto Hr. Sec. School - Madurai

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

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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

2
1
-2
0

\(\underset{x\rightarrow0}{lim}{\sqrt{1-cos 2x}\over x} \)

0
1
\(\sqrt{2}\)
does not exist

Choose the incorrect statement

log 1 to any base is zero
\(\frac{d}{dx}\) (e^x)=e^x
Inverse function of log x is \(\frac{1}{x}\)
|x| is not differentiable at x =0

At x\(={3\over 2}\) the function \(f(x)={|2x-3|\over 2x-3}\) is

continuous
discontinuous
differentiable
non-zero

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

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

\(\underset{x \rightarrow 0}{lim}{e^{tan \ x}-e^x\over tan x-x}=\)

1
e
\({1\over2}\)
0

If \(\underset{x \rightarrow 0}{lim}{sin \ px\over tan \ 3x}=4\) , then the value of p is

6
9
12
4

The value of \(\underset{x \rightarrow 0}{lim}{sin x\over \sqrt{x^2}}\) is

-1
1
0
limit does not exist

\(\underset{x \rightarrow 3}{lim}\left\lfloor x \right\rfloor =\)

2
3
does not exist
0

\(\underset{n \rightarrow \infty}{lim}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

\(1\over 2\)
0
1
\(\infty\)

Part - B

Generic 2 - Answer All Question

Find the points of discontinuity of the function f, where f(x)=\(\begin{cases}sinx, & \quad 0\le x \le \frac{\pi}{4} \\cosx, & \quad \frac{\pi}{4}<x< \frac{\pi}{2}\end{cases}\)

Evaluate the following limits :
\(\underset{x\rightarrow2}{lim}{x^4-16\over x-2}\)

Compute \(lim_{x\rightarrow1}{x^3-1\over x-1}\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{sin^3({x\over 2})\over x^3}\)
Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow{-3}}{lim}{\sqrt{1-x}-2\over x+3}\)

x -3.1 -3.01 -3.00 -2.999 -2.99 -2.9

f(x) 0.24845 -0.24984 -0.24998 -0.25001 -0.25015 -0.25158

Find \(lim_{x\rightarrow0}{(2+x)^5-2^5\over x}=lim_{y\rightarrow2}{y^5-2^5\over y-2}=5(2)^4=80\) .

Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow0}{lim}{cos x-1\over x}\)

x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) 0.04995 0.0049999 0.0004999 -0.0004999 -0.004999 -0.04995

Examine the continuity of the following:\(|x-2|\over |x+1|\)

Evaluate the following limits \(\underset{x\rightarrow\infty}{lim}{3\over x-2}-{2x+11\over x^2+x-6}\)

Suppose that the diameter of an animal’s pupils is given by \(f(x)={160x^{-0.4}+90\over 4x^{-0.4}+15},\) where x is the intensity of light and f(x) is in mm. Find the diameter of the pupils with maximum light.

Part - C

Generic 3 - Answer All Question

Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f. f(x)=\(\begin{cases} { (x-1) }^{ 3 }, & if\quad x<0 \\ { (x+1) }^{ 3 }, & if\quad x\ge 0 \end{cases}\)

Evaluate the following limits \(\underset{x\rightarrow\infty}{lim}({x^3\over2 x^2-1 }-{x^2\over 2x+1})\)

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

Evaluate \(\underset { x\rightarrow 0 }{ lim } f(x)\) ,where \(f(x)=\{ \begin{matrix} \frac { \left| x \right| }{ 0 } & x\neq 0 \\ 0 & x=0 \end{matrix}\)

Evaluate \(\lim _{ x\rightarrow \pi }{ \frac { \sin { x } }{ x-\pi } } \)

Show that \(\underset{x\rightarrow\infty}{lim}{1^2+2^2+....+(3n)^2\over (1+2+...+5)(2n+3)}={9\over25}\)

Compute \(lim_{x\rightarrow{0}}[{x^2+x\over x}+4x^3+3]\) .

Find the left and right limits of \(f(x)={x^2-4\over (x^2+4x+4)(x+3)}at \ x=-2\) .

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{3^x-1\over \sqrt{x+1}-1}\)
Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{1-cosx\over x^2}\)

Part - D

Generic 5 - Answer All Question

An important problem in fishery science is to estimate the number of fish presently spawning in streams and use this information to predict the number of mature fish or “recruits” that will return to the rivers during the reproductive period. If S is the number of spawners and R the number of recruits, “Beverton-Holt spawner recruit function”is \(R(S)={S\over (\alpha S+\beta)}\) where \(\alpha\) and \(\beta\) are positive constants. Show that this function predicts approximately constant recruitment when the number of spawners is sufficiently large.

Calculate \(lim_{x\rightarrow0}{1\over (x^2+x^3)}\)

Find the constant b that makes g continuous on \((-\infty,\infty)\) g(x) = \(\begin{cases} { x }^{ 2 }-{ b }^{ 2 } & if\ x<4 \\ bx+20 & if\ x\ge 4 \end{cases}\)

Evaluate \(\lim _{ x\rightarrow \frac { \pi }{ 2 } }{ \left( \frac { \pi }{ 2 } -x \right) \tan { x } } \)

Evaluate the following limits :
\(\underset{x\rightarrow5}{lim}{\sqrt{x-1}-2\over x-5}\)

Which of the following functions f has a removable discontinuity at x = x₀? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R
\(f(x)={x^3+64\over x+4},x_o=-4\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{e^{ax}-e^{bx}\over x}\)

Evaluate \(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ 4 }-1 }{ \sqrt { 1+x } -1 } } \)
Evaluate:\(\underset{x\rightarrow{3}}{lim}{x^2-9\over x-3}\) if it exists by finding f(3^-)and f(3⁺).

Show that \(lim_{x\rightarrow 0^+}x[\left\lfloor {1\over x} \right\rfloor+\left\lfloor {2\over x} \right\rfloor +....+\left\lfloor {15\over x}\right\rfloor ]=120\)

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