L'Hospital Rule to Remove Indeterminate Form

Q.

If $y={\log }_{\cos x}\sin x$, then $\frac{dy}{dx}$ is equal to

1. $\frac{[\cot x\mathrm{logcos}x+\tan x\mathrm{logsin}x)]}{(\mathrm{logcos}x{)}^2}$

2. $\frac{[\tan x\mathrm{logcos}x+\cot x\mathrm{logsin}x)]}{(\mathrm{logcos}x{)}^2}$

3. $\frac{[\cot x\mathrm{logcos}x+\tan x\mathrm{logsin}x)]}{(\mathrm{logsin}x{)}^2}$

4. none of these

Q.

The integral ${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{3\mathrm{\pi }}{4}}\frac{dx}{1+\cos x}$ is equal to

1. $2$

2. $4$

3. $-1$

4. $-2$

Q.

If $x$ is measured in degrees, then $\frac{d}{dx}(\cos x)$ is equal to

1. $-\sin x$

2. $-\left(\frac{180}{\mathrm{\pi }}\right)\sin x$

3. $-\left(\frac{\mathrm{\pi }}{180}\right)\sin x$

4. $\sin x$

Q.

The value of ${\int }_{-\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}{\sin }^{-4}\left(x\right)dx$ is

1. $-\frac{8}{3}$

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

3. $\frac{8}{3}$

4. none of these

Q.

The value of $\underset{x\to 0}{\lim }\frac{\left[{\int }_0^x\cos \left(t\right)dt\right]}{x}$ is

1. $1$

2. $-1$

3. $\infty$

4. None of these.

Q.

$\underset{x\to 0}{\lim }\frac{8\sin x+x\cos x}{3\tan x+x^2}=$

1. $3$

2. $2$

3. $-1$

4. $4$

Q.

Evaluate $\underset{x\to 0}{\lim }\frac{{\int }_0^{x}t\sin \left(10t\right)dt}{x}$ is equal to

1. $0$

2. $\frac{1}{10}$

3. $-\frac{1}{10}$

4. $-\frac{1}{5}$

Q.

For $0<x<2$, $\frac{d({\tan }^{-1}\sqrt{\left({\displaystyle \frac{1+\cos {\displaystyle \frac{1}{2}}}{1-\cos \frac{1}{2}}}\right)}}{dx}$is equal to

1. $\frac{-1}{4}$

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

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

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

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.

What is limit in basic calculus?

Q.

The integral ${\int }_0^{\mathrm{\pi }}{\left|\cos x\right|}^{3}dx$ is equal to;

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

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

3. $0$

4. $\frac{8}{3}$

Q.

If $y={\tan }^{-1}\left(2x\right)$, then $\frac{dy}{dx}$ is:

Q. $\underset{y\to 0}{lim}\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$

1. does not exist
2. exists and equals $\frac{1}{2\sqrt{2}}$
3. exists and equals $\frac{1}{4\sqrt{2}}$
4. exists and equals $\frac{1}{2\sqrt{2}(\sqrt{2}+1)}$

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. Let $f:(0,\pi )\to R$ be a twice differentiable function such that $\underset{t\to x}{lim}\frac{f\left(x\right)\sin t-f\left(t\right)\sin x}{t-x}={\sin }^{2}x$ for all $x\in (0,\pi )$.
If $f\left(\frac{\pi }{6}\right)=-\frac{\pi }{12}$, then which of the following statement(s) is (are) TRUE?

1. $f\left(\frac{\pi }{4}\right)=\frac{\pi }{4\sqrt{2}}$
2. $f\left(x\right)<\frac{x^4}{6}-x^2$ for all $x\in (0,\pi )$
3. There exists $\alpha \in (0,\pi )$ such that $f^{\prime }\left(\alpha \right)=0$
4. $f^{\prime \prime }\left(\frac{\pi }{2}\right)+f\left(\frac{\pi }{2}\right)=0$

Q. In $△ABC,\frac{sin(A-B)}{sin(A+B)}=$
[MP PET 1986]

1. $\frac{a^2-b^2}{c^2}$
2. $\frac{a^2+b^2}{c^2}$
3. $\frac{c^2}{a^2+b^2}$
4. $\frac{c^2}{a^2-b^2}$

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 $f\left(x\right)=\left[x\right]$ and $g\left(x\right)=$$\{\begin{array}{c}0,x\in Z\\ x^2,x\in R-Z\end{array}$
$\left(\right[.]$ represents greatest integer function$).$ Then

1. $f\left(x\right)$ is not continuous at $x=1.$
2. $gof$ is continuous for all $x.$
3. $fog$ is continuous for all $x.$
4. $\underset{x\to 1}{lim}g\left(x\right)$ exists but $g\left(x\right)$ is not continuous at $x=1.$

Q. If the range of the function $f\left(x\right)=8({\sin }^{4}x+{\cos }^{4}x-\sin x\cos x)\forall x\in R$ is $[a,b],$ then the value of ${\left(\underset{x\to a}{lim}\frac{(3x+b-1{)}^{1/3}-(b-1{)}^{1/3}}{x-a}\right)}^{-1}$ is

Q. $\underset{x\to 0}{lim}\frac{e^{x^2}-\cos x}{{\sin }^{2}x}$ is equal to:

1. $2$
2. $3$
3. $\frac{5}{4}$
4. $\frac{3}{2}$

Q. If $f\left(x\right)=\frac{1-\sin x}{\sin 2x},x\ne \frac{\pi }{2}$ is continuous at $x=\frac{\pi }{2},$ then the value of $f\left(\frac{\pi }{2}\right)$ is

1. $2$
2. $\frac{1}{2}$
3. $0$
4. $1$

Q.

$\underset{x\to 0}{lim}\frac{\sin 3x}{\sin 5x}$ is equal to

1. $3$

2. $5$

3. $\frac{3}{5}$

4. $\frac{5}{3}$

Q.

Let $f\left(a\right)=g\left(a\right)=k$ and their $n^{th}$ derivatives $f^n\left(a\right)$, $g^n\left(a\right)$ exist and are not equal for some $n$. Further if $\underset{x\to a}{lim}\frac{f\left(a\right)g\left(x\right)-f\left(a\right)-g\left(a\right)f\left(x\right)+g\left(a\right)}{g\left(x\right)-f\left(x\right)}=4$, then the value of $k$ is:

1. 4

2. 2

3. 1

4. 0

Q. $\underset{x\to 0}{lim}\frac{e^{x^2}-\cos x}{{\sin }^{2}x}$ is equal to:

1. $2$
2. $3$
3. $\frac{5}{4}$
4. $\frac{3}{2}$

Q. Let $f:R\to R$ be a differentiable function satisfying $f^{\prime }\left(3\right)+f^{\prime }\left(2\right)=0$. Then $\underset{x\to 0}{lim}{\left(\frac{1+f(3+x)-f\left(3\right)}{1+f(2-x)-f\left(2\right)}\right)}^{1/x}$ is equal to :

1. $e$
2. $1$
3. $e^{-1}$
4. $e^2$

Q. If $\underset{x\to 1}{lim}\frac{x^2-ax+b}{x-1}=5,$ then $a+b$ is equal to :

1. $1$
2. $5$
3. $-4$
4. $-7$

Q. If $f:R\to R$ is a differentiable function and $f\left(2\right)=6$, then $\underset{x\to 2}{lim}\underset{6}{\overset{f\left(x\right)}{\int }}\frac{2tdt}{(x-2)}$ is :

1. $0$
2. $2f^{\prime }\left(2\right)$
3. $12f^{\prime }\left(2\right)$
4. $24f^{\prime }\left(2\right)$

Q. If $\underset{x\to 1}{lim}\frac{x^2-ax+b}{x-1}=5,$ then $a+b$ is equal to :

1. $1$
2. $5$
3. $-4$
4. $-7$

Q. If $\alpha =\underset{n\to \infty }{lim}(\frac{1}{n^3+1}+\frac{4}{n^3+1}+\frac{9}{n^3+1}+\cdots +\frac{n^2}{n^3+1})$ and $\beta =\underset{x\to 0}{lim}\frac{\sin 2x}{\sin 8x},$ then a quadratic equation whose roots are $\alpha$ and $\beta$ is

1. $12x^2-7x+1=0$
2. $x^2+19x-120=0$
3. $x^2-17x+66=0$
4. $x^2-7x+12=0$