L'Hospital Rule to Remove Indeterminate Form

Q.

The largest value of non-negative integer $a$ for which
${lim}_{x\to 1}{\left\{\frac{-ax+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$ is ___

Q. Let $p={lim}_{x\to 0^{+}}(1+ta{n}^2\sqrt{x}{)}^{\frac{1}{2x}}$ then log p is equal to:

1. $\frac{1}{2}$
2. $\frac{1}{4}$
3. $2$
4. $1$

Q.

If $0<a,b<1$, and ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\mathrm{\pi }}{4}$, then the value of $\left(a+b\right)-\frac{a^2+b^2}{2}+\frac{a^3+b^3}{3}-\frac{a^4+b^4}{4}+...$ is

1. ${\log }_e\left(2\right)$

2. ${\log }_e\left(\frac{e}{2}\right)$

3. $e$

4. $e^2-1$

Q.

If $y=\log \tan \sqrt{x}$, then the value of $\frac{dy}{dx}$is:

1. $\frac{1}{2\sqrt{x}}$

2. $\frac{{\sec }^2\sqrt{x}}{\left(\sqrt{x}\tan \sqrt{x}\right)}$

3. $2{\sec }^2\sqrt{x}$

4. $\frac{{\sec }^2\sqrt{x}}{\left(2\sqrt{x}\tan \sqrt{x}\right)}$

Q. Let $f:R\to (0,\infty )$ and $g:R\to R$ be twice differentiable functions such that $f^{\prime \prime }$ and $g^{\prime \prime }$ are continuous functions on $R$. Suppose $f^{\prime }\left(2\right)=g\left(2\right)=0,f^{\prime \prime }\left(2\right)\ne 0$ and $g^{\prime }\left(2\right)\ne 0$. If $\underset{x\to 2}{lim}\frac{f\left(x\right)g\left(x\right)}{f^{\prime }\left(x\right)g^{\prime }\left(x\right)}=1,$ then

1. $f$ has a local minimum at $x=2$
2. $f$ has a local maximum at $x=2$
3. $f^{\prime \prime }\left(2\right)>f\left(2\right)$
4. $f\left(x\right)-f^{\prime \prime }\left(x\right)=0$ for at least one $x\in R$

Q.

If $li{m}_{x\to 2}$ $\frac{\left(x^n\right)-\left(2^n\right)}{x-2}$ =80 , where n is a positive integer,
then n=

1. 3

2. 2

3. 5

4. None of these

Q.

Evaluate: $\underset{x\to 0}{lim}\frac{cos2x-1}{cosx-1}$

Q.

Evaluate $\underset{x\to 2}{lim}\frac{x^{10}-1024}{x^5-32}$

1. 16

2. 64

3. 32

4. 128

Q.

$\text{The value of}\underset{x\to 1}{lim}\frac{x^n+x^{n-1}+x^{n-2}+.......+x^2+x-n}{x-1}$

1. 0

2. 1

3. n

4. 

Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{sin ax}}{\text{sin bx}},a,b\ne 0$

Q.

Evaluate $\underset{x\to 0}{lim}\frac{(1-cosx)}{x^2}$

Q.

How do you verify $\cos \left(180°\right)=-1$?

Q.

If sin(A+B+C)=1, tan(A-B)=$\frac{1}{\sqrt{3}}$ and Sec(A+C)=2, then

1. None of the above

2. 

3. 

4. 

Q.

$\text{The value of}\underset{x\to 1}{lim}\frac{x^n+x^{n-1}+x^{n-2}+.......+x^2+x-n}{x-1}$

1. $\frac{n(n+1)}{2}$

2. 0

3. 1

4. n

Q.

Let f(x) = 4 and f’(x) = 4. Then $\underset{x\to 0}{lim}\frac{xf\left(2\right)-2f\left(x\right)}{x-2}$ is given by

1. 2

2. -2

3. -4

4. 3

Q.

If $Li{m}_{x\to 0}\frac{4+sin2x+Asinx+Bcosx}{x^2}$ exists, then the values A and B are

1. -2 and -4

2. -4 and -2

3. -3 and -2

4. -1 and -4

Q.

If sinA+sinB=C, cosA+cosB=D, then the value of sin(A+B)=

[MP PET 1986]

1. CD

2. 

3. 

4. 

Q. If sin B= $\frac{1}{5}sin(2A+B),then\frac{tan(A+B)}{tanA}$ is equal to

1. 
2. 
3. 
4. 

Q.

$\underset{x\to 2}{lim}\frac{sin(e^{x-2}-1)}{log(x-1)}$

Q.

$\underset{x\to 0}{lim}{\left(\frac{sinx}{x}\right)}^{\left(\frac{sinx}{x-sinx}\right)}$ equals

1. e^-2

2. e^-1

3. 1

4. e

Q.

In a triangle ABC, if sin $AsinB=\frac{ab}{c^2},$ then the triangle is

1. isosceles

2. right angled

3. obtuse angled

4. equilateral

Q.

$\underset{x\to 3}{lim}\left(\right[x-3]+[3-x]-x),\text{where[.]denote the greatest function, is equal to:}$

1. 4

2. -4

3. 0

4. Does not exist

Q.

Evaluate $\underset{x\to 0}{lim}\frac{sinax}{bx}$

1. b

2. 

3. 

4. d

Q.

If $li{m}_{x\to 2}$ $\frac{\left(x^n\right)-\left(2^n\right)}{x-2}$ =80 , where n is a positive integer,
then n=

1. 3

2. 5

3. 2

4. None of these

Q.

$In\Delta ABC,(a-b{)}^{2}co{s}^2\frac{C}{2}+(a+b{)}^{2}si{n}^2\frac{C}{2}\text{is equal to}$

1. a²

2. b²

3. c²

4. 2a²

Q.

Let $f\left(x\right)$ be a twice differentiable function and $f^{\prime \prime }\left(0\right)=5$, then $\underset{x\to 0}{lim}\frac{3f\left(x\right)-4f\left(3x\right)+f\left(9x\right)}{x^2}$ is equal to:

1. 30

2. 120

3. 40

4. 5

Q. Evaluate the limit: $\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}a,b,a+b\ne 0$

Q.

${lim}_{x\to 0}\frac{ta{n}^{-1}x-si{n}^{-1}x}{x^3}\text{is equal to}$

1. 1/2

2. -1/2

3. 1

4. -1

Q. If y = sin (sin x) and $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}$ tan x+f(x) = 0, then f(x) equals.

1. 
2. 
3. 
4. 

Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{(x+1{)}^5-1}{x}$