Question

The function $‘f’$ is defined by $f (x) = 2x - 1$, if $x > 2, f (x) = k$ if $ x = 2$ and $x^2 - 1$, if $x < 2$ is continuous, then the value of k is equal to :

• A. 2
• B. 3
• C. 4
• D. -3

Answer 1

Answer: (B)

Let f is defined as $f (x) = 2x - 1$ if $ x > 2$
$= k$ if $x = 2$
$= x^2 - 1$ if $x < 2$
Since, f (x) is continuous.
$\therefore $ Limit of f (x) at x = 2 = value of f (x) at 2.
i.e., $\displaystyle\lim_{x \to 2} f(x) = f(2)$
Now, $\displaystyle\lim_{x \to 2} f(x) = \displaystyle\lim_{x \to 2} (2x - 1) = 3 = f(2) $
But given $f (x) = k$ at $x = 2$
$\therefore \:\: k = 3$