The value of $k$ for which the function, defined by, $f(x) = \left\{\begin{matrix}\frac{3x+4\tan x}{x}:&x≠0\\k:&x=0\end{matrix}\right.$ is continuous at $x = 0$, is

7

The correct answer is Option (3) → 7

Given:

$f(x) = \begin{cases} \frac{3x + 4\tan x}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$

To ensure continuity at $x = 0$, the following must hold:

$\lim\limits_{x \to 0} f(x) = f(0) = k$

Evaluate the limit:

$\lim\limits_{x \to 0} \frac{3x + 4\tan x}{x}$

$= \lim\limits_{x \to 0} \left( 3 + 4 \cdot \frac{\tan x}{x} \right)$

Using identity: $\lim\limits_{x \to 0} \frac{\tan x}{x} = 1$

$= 3 + 4 \cdot 1 = 7$

Hence, for continuity:

$k = 7$