L'Hospital Rule to Remove Indeterminate Form

Q.

Explain $\frac{u}{v}$ rule of differentiation.

Q.

${lim}_{x\to \infty }(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+....+\frac{1}{3^n})$

Q.

${lim}_{n\to \infty }(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+....+\frac{n-1}{n^2})$

Q. If $L=\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x^2}$, then the value of $\frac{1}{\ln L}$ is

Q. If $\underset{x\to 0}{lim}\frac{ax-(e^{4x}-1)}{ax(e^{4x}-1)}$ exists and is equal to $b,$ then the value of $a–2b$ is

Q.

Evaluate : ${\int }_0^{\pi }\frac{1}{\left[1+\sin x\right]}dx$

1. $1$

2. $2$

3. $-1$

4. $-2$

Q.

If $\tan \left(\alpha \right)=\frac{1}{7},\tan \left(\beta \right)=\frac{1}{3}$, then $\cos \left(2\alpha \right)=$

1. $\sin \left(2\beta \right)$

2. $\sin \left(4\beta \right)$

3. $\sin \left(3\beta \right)$

4. None of these

Q.

If $\alpha ={\cos }^{-1}\left(\frac{3}{5}\right),\beta ={\tan }^{-1}\left(\frac{1}{3}\right),$ where $0<\alpha ,\beta <\frac{\pi }{2}$, then $\alpha -\beta$ is equal to:

1. ${\tan }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

2. ${\sin }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

3. ${\tan }^{-1}\left(\frac{9}{15}\right)$

4. ${\cos }^{-1}\left(\frac{9}{5\sqrt{10}}\right)$

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.

How do you simplify the expression $co{t}^{2}x+1$?

Q.

$\int e^{x\log a}{e}^{x}dx=$

1. $\left(\frac{a^x}{\log ae}\right)+c$

2. $\left(\frac{e^x}{1+{\log }_{e}a}\right)+c$

3. ${\left(ae\right)}^x+c$

4. $\left(\frac{a{e}^x}{{\log }_{e}ae}\right)+c$

5. $\left(\frac{a^x{e}^x}{{\log }_{x}e}\right)+c$

Q. $\underset{x\to 0}{lim}\frac{x+2\sin x}{\sqrt{x^2+2\sin x+1}-\sqrt{{\sin }^{2}x-x+1}}=$

1. $1$
2. $2$
3. $3$
4. $6$

Q.

The number of values of $x$ in the interval $[0,3\pi ]$ satisfying the equation $2{\sin }^{2}x+5\sin x-3=0$ is

1. $6$

2. $1$

3. $2$

4. $4$

Q. Statement $1:$ $f\left(x\right)=\sin x+\left[x\right]$ is discontinuous at $x=0,$ where $[.]$ denotes the greatest integer function.
Statement $2:$ If $g\left(x\right)$ is continuous and $h\left(x\right)$ is discontinuous at $x=a,$ then $g\left(x\right)+h\left(x\right)$ will necessarily be discontinuous at $x=a.$

1. Both the statements are true and statement $2$ is the correct explanation of statement$1$.
2. Both the statements are true and statement $2$ is not the correct explanation of statement $1$.
3. statement $1$ is true andstatement $2$ is false
4. statement $1$ is false and statement $2$ is true

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

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

Q.

The value of $\frac{dy}{dx}$ at $x=\frac{\pi }{2}$, where $y$ is given by $y=x^{\sin x}+\sqrt{x}$, is

1. $1+\frac{1}{\sqrt{2\mathrm{\pi }}}$

2. $1$

3. $\frac{1}{\sqrt{2\mathrm{\pi }}}$

4. $1-\frac{1}{\sqrt{2\mathrm{\pi }}}$

Q. Given that for each $a\in (0,1),$
$\underset{h\to 0^{+}}{lim}\underset{h}{\overset{1-h}{\int }}{t}^{-a}(1-t{)}^{a-1}dt$
exists. Let this limit be $g\left(a\right)$. In addition, it is given that the function $g\left(a\right)$ is differentiable on $(0,1)$.

The value of $g^{\prime }\left(\frac{1}{2}\right)$ is

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

Q. Evaluate the limit:
$\underset{x\to 0}{lim}\frac{\sin 3x+7x}{4x+\sin 2x}$

Q.

What is $\sin \left(4x\right)$ in terms of $\sin \left(2x\right)$ and $\cos 2x$?

Q.

The differential coefficient of the given function ${\log }_e\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$is

1. $\cos ecx$

2. $\sec x$

3. $\tan x$

4. $\cos x$

Q.

${lim}_{n\to \infty }\frac{1^3+2^3+3^3+....+n^3}{n^4}$

Q. If $\underset{x\to 0}{lim}\frac{1}{x^8}[1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+\cos \frac{x^2}{2}\cos \frac{x^2}{4}]=2^{-k},$ then the value of $k$ is​​​​​​​

Q. Let $f:R\to R$ be a continuously differentiable function such that $f\left(2\right)=6$ and $f^{\prime }\left(2\right)=\frac{1}{48}.$
If $\underset{6}{\overset{f\left(x\right)}{\int }}4t^{3}dt=(x-2)g\left(x\right),$ then $\underset{x\to 2}{lim}g\left(x\right)$ is equal to

1. $12$
2. $18$
3. $24$
4. $36$

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

Q.

$\underset{x\to 0}{\lim }\frac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}=$

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

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

3. $1$

4. $2$

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

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

Q.

Evaluate:

${lim}_{n\to \infty }\frac{1.2+2.3+3.4+...+n(n+1)}{n^3}$

Q.

Evaluate $\underset{x\to 0}{lim}\frac{(x+1{)}^5-1}{x}$

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Q.

The value of $a$ in order that $f\left(x\right)=\sin x-\cos x-ax+b$ decreases for all real value of $x$ is given by

1. $a\ge \sqrt{2}$

2. $a<\sqrt{2}$

3. $a\ge 1$

4. $a<1$

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. $f^{\prime \prime }\left(\frac{\pi }{2}\right)+f\left(\frac{\pi }{2}\right)=0$