Question

If $f(x) = \left\{ \begin{matrix} 2x + 1\text{when}x < 1 \\ k\text{when}x = 1 \\ 5x - 2\text{when}x > 1 \end{matrix} \right.\$ is continuous at x =1, then the value of k is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

Since $f(x)$ is continuous at x = 1,

⇒ $\lim_{x \rightarrow 1^{-}}f(x) = \lim_{x \rightarrow 1^{+}}f(x) = f(1)$ …..(i)

Now $\lim_{x \rightarrow 1^{-}}f(x) = \lim_{h \rightarrow 0}f(1 - h)$ = $\lim_{h \rightarrow 0}2(1 - h) + 1 = 3$ i.e.,

$\lim_{x \rightarrow 1^{-}}f(x) = 3$

Similarly, $\lim_{x \rightarrow 1^{+}}f(x) = \lim_{h \rightarrow 0}f(1 + h)$ =$\lim_{h \rightarrow 0}5(1 + h) - 2$ i.e.,

$\lim_{x \rightarrow 1^{+}}f(x) = 3$

So according to equation (i), we have k = 3.