Question

If $f(x) = \frac{\tan(3x)}{x} + \frac{x}{2}$ is continuous at $x = 0$, then $f(0)$ is equal to

• A. $3$
• B. $2$
• C. $4$
• D. $0$

Hint:

$\lim_{x\to 0} \frac{\tan ax}{x} = a$.

Answer 1

The Correct Option is C

Step 1: $\lim_{x\to 0} \frac{\tan 3x}{x} = \lim_{x\to 0} \frac{3x}{x} = 3$.}
Step 2: $\lim_{x\to 0} f(x) = 3 + 0 = 3$. For continuity, $f(0)=3$. But options have 4? $f(x) = \frac{\tan 3x}{x} + \frac{x}{2}$, at $x=0$, limit = 3, so $f(0)=3$ not in options. Possibly $f(x)=\frac{\tan 3x}{x} + \frac{x}{2}$ has limit 3. So answer is 3.}