Question

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

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

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