Consider the following in respect of the function \[f\left( x \right) = \left\{ \begin{array}{l}
2 + x,\,\,\,\,\,x \ge 0\\
2 - x,\,\,\,\,\,x < 0
\end{array} \right.\]
$$\eqalign{
& 1.\,\,\mathop {\lim }\limits_{x \to 1} f\left( x \right)\,{\text{does not exist}} \cr
& 2.{\text{ }}\,f\left( x \right)\,{\text{is differentiable at }}x = 0 \cr
& 3.{\text{ }}\,f\left( x \right)\,{\text{is continuous at }}x = 0 \cr} $$
Which of the above statements is/are correct ?
Releted MCQ Question on Calculus >> Differentiability and Differentiation
Releted Question 1
There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-
A.
$$f''\left( x \right) > 0$$ for all $$x$$
B.
$$ - 1 < f''\left( x \right) < 0$$ for all $$x$$
C.
$$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$
If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-
Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-