MIDTERM EXAM

December $4,2021$

$4.$ Let $f(x)=\{\begin{array}{cc}{x}^2+ax+b,& x<mn>1</mn>\\ 1,& x=1\\ -1+(x-1)sin(\frac{1}{x-1}),& x>1\end{array}$

Which of the following is true?

$($a$)$ If $\underset{x1}{lim}f(x)$ exists, $a+b=1.$

$($b$)$ If $\underset{x1}{lim}f(x)$ exists, $a+b=0.$

$($c$)$ If $\underset{x1}{lim}f(x)$ exists, $a+b=-2.$

$($c$)$ If $\underset{x1}{lim}f(x)$ exists, $a+b=-1.$

$($e$)\underset{x1}{lim}f(x)$ does not exist for any value of a and b$.$