Limits, Continuity & Differentiability · JEE Main previous year questions

Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime }(x)\ge 2$ for $x\in [1,6]$, then

1. A.$f(6)\ge 8$
2. B.$f(6)<8$
3. C.$f(6)<5$
4. D.$f(6)=5$

If ${\lim }_{x\to \infty }{\left(1+\frac{a}{x}+\frac{b}{x^2}\right)}^{2x}=e^2$, then the value of $a$ and $b$, are

1. A.$a=1$ and $b=2$
2. B.$a=1$ and $b\in R$
3. C.$a\in R$ and $b=2$
4. D.$a\in R$ and $b\in R$

Let $f(x)=\frac{1-\tan x}{4x-\pi },x\ne \frac{\pi }{4},x\in \left[0,\frac{\pi }{2}\right]$. If $f(x)$ is continuous in $\left[0,\frac{\pi }{2}\right]$, then $f\left(\frac{\pi }{4}\right)$ is

1. A.$-1$
2. B.$\frac{1}{2}$
3. C.$-\frac{1}{2}$
4. D.$1$

${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{1}{n}{e}^{\frac{r}{n}}$ is

1. A.$e+1$
2. B.$e-1$
3. C.$1-e$
4. D.$e$

${\lim }_{x\to \infty }{\left(\frac{x^2+5x+3}{x^2+x+2}\right)}^x$

1. A.$e^4$
2. B.$e^2$
3. C.$e^3$
4. D.$1$

Let $f(2)=4$ and $f^{\prime }(x)=4$. Then ${\lim }_{x\to 2}\frac{xf(2)-2f(x)}{x-2}$ is given by

1. A.$2$
2. B.$-2$
3. C.$-4$
4. D.$3$

${\lim }_{x\to 0}\frac{\log x^n-[x]}{[x]},n\in N$, ( [x] denotes the greatest integer less than or equal to x )

1. A.has value $-1$
2. B.has value $0$
3. C.has value $1$
4. D.does not exist

If $f(1)=1,f^{\prime }(1)=2$, then ${\lim }_{x\to 1}\frac{\sqrt{f(x)}-1}{\sqrt{x}-1}$ is

1. A.$2$
2. B.$4$
3. C.$1$
4. D.$\frac{1}{2}$