Question
Let \[f\left( x \right) = \left\{ \begin{array}{l}
3x - 4,\,\,\,\,\,0 \le x \le 2\\
2x + \ell ,\,\,\,\,\,\,2 < x \le 9
\end{array} \right.\]
If $$f$$ is continuous at $$x = 2,$$ then what is the value of $$\ell \,?$$
A.
$$0$$
B.
$$2$$
C.
$$ - 2$$
D.
$$ - 1$$
Answer :
$$ - 2$$
Solution :
Given function is : \[f\left( x \right) = \left\{ \begin{array}{l}
3x - 4,\,\,\,\,\,0 \le x \le 2\\
2x + \ell ,\,\,\,\,\,\,2 < x \le 9
\end{array} \right.\] and also given that $$f\left( x \right)$$ is continuous at $$x = 2$$
For a function to be continuous at a point $${\text{L}}{\text{.H}}{\text{.L}}{\text{.}} = {\text{R}}{\text{.H}}{\text{.L}}{\text{.}} = {\text{V}}{\text{.F}}{\text{.}}$$ at that point. $$f\left( 2 \right) = 2 = {\text{V}}{\text{.F}}{\text{.}}$$
$$\eqalign{
& \Rightarrow {\text{R}}{\text{.H}}{\text{.L}}{\text{.}}\,:\,\mathop {\lim }\limits_{x \to 2} \left( {2x + \ell } \right) = 3\left( 2 \right) - 4 \cr
& \Rightarrow \mathop {\lim }\limits_{h \to 0} \left\{ {2\left( {2 + h} \right) + \ell } \right\} = 6 - 4 \cr
& \Rightarrow 4 + \ell = 2 \cr
& \Rightarrow \ell = - 2 \cr} $$