L'Hospital Rule to Remove Indeterminate Form

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.

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.

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.

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.

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

1. 0

2. 1

3. n

4. 

Q. $\underset{x\to 0}{lim}\frac{(1+x{)}^5-1}{3x+5x^2}$ is equal to

1. 
2. 
3. 
4. 

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. The value of ${lim}_{x\to 0}\left(\right(sinx{)}^{\frac{1}{x}}+\left(1/x{)}^{sinx}\right)$ where $x>0$ is

1. 0
2. -1
3. 1
4. 2

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.

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

1. -1/2

2. 1/2

3. 1

4. -1