Question
Consider the function \[f\left( x \right) = \left\{ \begin{array}{l}
\,\,\,\,\,\,\,\,ax - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for}}\,\,\,\, - 2 < x < 1\\
\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for}}\,\,\,\, - 1 \le x \le 1\\
a + 2{\left( {x - 1} \right)^2}\,\,\,\,\,\,\,{\rm{for}}\,\,\,\,\,\,\,\,\,1 < x < 2
\end{array} \right.\]
What is the value of a for which $$f\left( x \right)$$ is continuous at $$x = - 1$$ and $$x = 1?$$
A.
$$ - 1$$
B.
$$1$$
C.
$$0$$
D.
$$2$$
Answer :
$$ - 1$$
Solution :
\[f\left( x \right) = \left\{ \begin{array}{l}
\,\,\,\,\,\,\,\,ax - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 2 < x < 1\\
\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 1 \le x \le 1\\
a + 2{\left( {x - 1} \right)^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 < x < 2
\end{array} \right.\]
$$\eqalign{
& {\text{if }}f\left( x \right){\text{ is continuous at }}x = - 1 \cr
& {\text{then, }}\mathop {\lim }\limits_{x \to - 1} \left( {ax - 2} \right) = \mathop {\lim }\limits_{x \to - 1} \left( { - 1} \right) \cr
& \Rightarrow a\left( { - 1} \right) - 2 = - 1 \cr
& \Rightarrow a = - 1 \cr
& {\text{if }}f\left( x \right){\text{ is continuous at }}x = 1 \cr
& {\text{then, }}\mathop {\lim }\limits_{x \to 1} a + 2{\left( {x - 1} \right)^2} = \mathop {\lim }\limits_{x \to 1} \left( { - 1} \right) \cr
& \Rightarrow a + 2{\left( {1 - 1} \right)^2} = - 1 \cr
& \Rightarrow a = - 1 \cr} $$