65
MCQ (Single Correct Answer)
If $f(x)=\left\{\begin{array}{ll}x^2-1, & 0< x< 2 \\ 2 x+3, & 2 \leq x < 3\end{array}\right.$, then the quadratic equation whose roots are $\lim _\limits{x \rightarrow 2^{-}} f(x)$ and $\lim_\limits{x \rightarrow 2^{+}}{f(x)}$ is
A
$x^2-6 x+9=0$
B
$x^2-7 x+8=0$
C
$x^2-14 x+49=0$
D
$x^2-10 x+21=0$
66
MCQ (Single Correct Answer)
$\lim _{x \rightarrow 0} \frac{\tan 2 x-x}{3 x-\sin x}$ is equal to
A
2
B
$\frac{1}{2}$
C
$\frac{-1}{2}$
D
$\frac{1}{4}$
67
MCQ (Single Correct Answer)
If $f(x)=x-[x], \in R$, then $f^{\prime}\left(\frac{1}{2}\right)$ is equal to
A
$\frac{3}{2}$
B
1
C
0
D
$-1$
68
MCQ (Single Correct Answer)
If $y=\sqrt{x}+\frac{1}{\sqrt{x}}$, then $\frac{d y}{d x}$ at $x=1$ is equal to
A
1
B
$\frac{1}{2}$
C
$\frac{1}{\sqrt{2}}$
D
0
69
MCQ (Single Correct Answer)
If $f(x)=\frac{x-4}{2 \sqrt{x}}$, then $f^{\prime}(1)$ is equal to
A
$\frac{5}{4}$
B
$\frac{4}{5}$
C
1
D
0