11th Maths Monthly Test - 3( Differential Calculus - Limits and Continuity )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3( Differential Calculus - Limits and Continuity ) -Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow{0}}{lim}\ {\sqrt{x+3}-\sqrt{3}\over x}\)

x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) 0.2911 0.2891 0.2886 0.2886 0.2885 0.28631

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

Examine the continuity of the following: e^2x + x²

Calculate \(lim_{x\rightarrow x_o}(5)\) for any real number x₀.

Discuss the continuity at x = 0 for \(f\left( x \right) =\begin{cases} \frac { 1-\cos { x } }{ { x }^{ 2 } } ,\quad x\neq 0 \\ \frac { 1 }{ 2 } ,\quad \quad \quad x=0 \end{cases}\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}(1+x)^{1\over3x} \)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { a }^{ x }-1 }{ { b }^{ x }-1 } } \)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ \frac { 2 }{ 3 } }-9 }{ x-27 } } \)

Show that the function is \(f\left( x \right) =\begin{cases} \frac { \sin { x } }{ x } +\cos { x,\quad x\neq 0 } \\ 2,\quad \quad \quad x=0 \end{cases}\) continuous at x =0.

Evaluate the following limits :
\(\underset{x\rightarrow5}{lim}{\sqrt {x+4}-3\over x-5}\)

Prove that f(x) = 2x² + 3x - 5 is continuous at all points in R.

Compute \(lim_{x\rightarrow8}(5x)\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{sin \alpha x\over sin \beta x}\)

Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow 2}{lim}\ {x-2\over x^2-x-2}\)

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x) 0.344820 0.33444 0.33344 0.3333222 0.33222 0.332258

Part - B

Generic 3 - Answer All Question

If f and g are continuous functions with f(3) = 5 and \(\underset{x\rightarrow3}{lim}[2f(x)-g(x)]=4\) ,find g(3).

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{tan 2x\over x}\)

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

Calculate \(lim_{x\rightarrow \infty}{x^3+2x+3\over (5x^2+1)}\) .

Find the left and right limits of f(x) = tanx at x = \(\frac { \pi }{ 2 } \)

Evaluate \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 3 }-8 }{ { x }^{ 2 }-4 } } \)

Alcohol is removed from the body by the lungs, the kidneys, and by chemical processes in liver. At moderate concentration levels, the majority work of removing the alcohol is done by the liver; less than 5% of the alcohol is eliminated by the lungs and kidneys. The rate r at which the liver processes alcohol from the bloodstream is related to the blood alcohol concentration x by a rational function of the form \(r(x)={\alpha x\over x+\beta}\) for some positive constants \(\alpha\) and \(\beta\). Find the maximum possible rate of removal.

\(It\quad \lim _{ x\rightarrow a }{ \frac { { x }^{ 9 }-{ a }^{ 9 } }{ x-a } } =9,\)find all possible values of a.

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

Show that \(\underset{x\rightarrow\infty}{lim} {1\over 1.2}+{1\over 2.3}+{1\over 3.4}+...+{1\over n(n+1)}=1\)

Calculate \(lim_{x\rightarrow -1}(x^2-3)^{10}\)

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin(x -\left\lfloor x \right\rfloor) \over x- \left\lfloor x \right\rfloor\)

Find \(lim_{t\rightarrow0}{\sqrt{t^2+9}-3\over t^2}.\)

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

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x	-0.1	-0.01	-0.001	0.001	0.01	0.1
f(x)	0.2911	0.2891	0.2886	0.2886	0.2885	0.28631

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	0.344820	0.33444	0.33344	0.3333222	0.33222	0.332258

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11th Maths Monthly Test - 3( Differential Calculus - Limits and Continuity )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3( Differential Calculus - Limits and Continuity ) -Aug 2020

Part - A

Generic 2 - Answer All Question

Complete the table using calculator and use the result to estimate the limit. \(\underset{x\rightarrow{0}}{lim}\ {\sqrt{x+3}-\sqrt{3}\over x}\) x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) 0.2911 0.2891 0.2886 0.2886 0.2885 0.28631

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

Examine the continuity of the following: e2x + x2

Calculate \(lim_{x\rightarrow x_o}(5)\) for any real number x0.

Discuss the continuity at x = 0 for \(f\left( x \right) =\begin{cases} \frac { 1-\cos { x } }{ { x }^{ 2 } } ,\quad x\neq 0 \\ \frac { 1 }{ 2 } ,\quad \quad \quad x=0 \end{cases}\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}(1+x)^{1\over3x} \)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { a }^{ x }-1 }{ { b }^{ x }-1 } } \)

Evaluate\(\lim _{ x\rightarrow 0 }{ \frac { { x }^{ \frac { 2 }{ 3 } }-9 }{ x-27 } } \)

Show that the function is \(f\left( x \right) =\begin{cases} \frac { \sin { x } }{ x } +\cos { x,\quad x\neq 0 } \\ 2,\quad \quad \quad x=0 \end{cases}\) continuous at x =0.

Evaluate the following limits : \(\underset{x\rightarrow5}{lim}{\sqrt {x+4}-3\over x-5}\)

Prove that f(x) = 2x2 + 3x - 5 is continuous at all points in R.

Compute \(lim_{x\rightarrow8}(5x)\)

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{sin \alpha x\over sin \beta x}\)

Complete the table using calculator and use the result to estimate the limit. \(\underset{x\rightarrow 2}{lim}\ {x-2\over x^2-x-2}\) x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) 0.344820 0.33444 0.33344 0.3333222 0.33222 0.332258

Part - B

Generic 3 - Answer All Question

If f and g are continuous functions with f(3) = 5 and \(\underset{x\rightarrow3}{lim}[2f(x)-g(x)]=4\) ,find g(3).

Evaluate the following limits :\(\underset{x\rightarrow 0}{lim}{tan 2x\over x}\)

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

Calculate \(lim_{x\rightarrow \infty}{x^3+2x+3\over (5x^2+1)}\) .

Find the left and right limits of f(x) = tanx at x = \(\frac { \pi }{ 2 } \)

Evaluate \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 3 }-8 }{ { x }^{ 2 }-4 } } \)

\(It\quad \lim _{ x\rightarrow a }{ \frac { { x }^{ 9 }-{ a }^{ 9 } }{ x-a } } =9,\)find all possible values of a.

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

Show that \(\underset{x\rightarrow\infty}{lim} {1\over 1.2}+{1\over 2.3}+{1\over 3.4}+...+{1\over n(n+1)}=1\)

Calculate \(lim_{x\rightarrow -1}(x^2-3)^{10}\)

Do the limits of following functions exist as x\(\rightarrow 0?\) State reasons for your answer.\(sin(x -\left\lfloor x \right\rfloor) \over x- \left\lfloor x \right\rfloor\)

Find \(lim_{t\rightarrow0}{\sqrt{t^2+9}-3\over t^2}.\)

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

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Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow{0}}{lim}\ {\sqrt{x+3}-\sqrt{3}\over x}\)

x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) 0.2911 0.2891 0.2886 0.2886 0.2885 0.28631

Examine the continuity of the following: e^2x + x²

Calculate \(lim_{x\rightarrow x_o}(5)\) for any real number x₀.

Evaluate the following limits :
\(\underset{x\rightarrow5}{lim}{\sqrt {x+4}-3\over x-5}\)

Prove that f(x) = 2x² + 3x - 5 is continuous at all points in R.

Complete the table using calculator and use the result to estimate the limit.
\(\underset{x\rightarrow 2}{lim}\ {x-2\over x^2-x-2}\)

x 1.9 1.99 1.999 2.001 2.01 2.1

f(x) 0.344820 0.33444 0.33344 0.3333222 0.33222 0.332258

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