Question

If the function $f(x)=\begin{cases}x^{2}, \text { if } x \leq 4 \\ a x, \text { if } x>4\end{cases}.$
is continuous at $x=4$, then $a=$

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

$L.H.L =\displaystyle\lim _{x \rightarrow 4^{-}} f(x)=\displaystyle\lim _{x \rightarrow 4^{-}} x^{2}$
Put $x=4- h .$ as $x \rightarrow 4, h \rightarrow 0$
$\therefore \,\,\, L.H.L. =\displaystyle\lim _{ h \rightarrow 0}(4- h )^{2}=\displaystyle\lim _{ h \rightarrow 0}\left(16+ h ^{2}-8 h \right)=16$
$R . H.L =\displaystyle\lim _{ x \rightarrow 4^{+}} f ( x )=\displaystyle\lim _{ x \rightarrow 4^{+}} ax$
Put $x=4+ h .$ as $x \rightarrow 4, h \rightarrow 0$
$\therefore \,\, R.H.L. =\displaystyle\lim _{ h \rightarrow 0} a (4+ h )=\displaystyle\lim _{ h \rightarrow 0} 4 a + ah =4 a$
For the function to be continuous at $x =4, L . H . L$.
$= R . H . L .$ Therefore $16=4 a$ which gives $a =4 .$