Question

If 

then the value of \( m \) is: 
 

• A. \( \frac{5}{8} \)
• B. \( \frac{8}{5} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{5}{3} \)

Hint:

For a function to be continuous at a point, the limit of the function at that point must equal the value of the function at that point.

Answer 1

The Correct Option is A

Step 1: Continuity Condition. 
For the function to be continuous at \( x = 0 \), we must have: \[ \lim_{x \to 0} f(x) = f(0) \] Thus, we need to calculate \( \lim_{x \to 0} \frac{\sin 8x}{5x} \). 
Step 2: Finding the limit. 
Using the standard limit \( \lim_{x \to 0} \frac{\sin kx}{x} = k \), we get: \[ \lim_{x \to 0} \frac{\sin 8x}{5x} = \frac{8}{5} \] Step 3: Conclusion. 
For the function to be continuous, we must have: \[ f(0) = m + 1 = \frac{8}{5} \] Thus, solving for \( m \), we get: \[ m + 1 = \frac{8}{5} \quad \Rightarrow \quad m = \frac{8}{5} - 1 = \frac{5}{8} \] Step 4: Final Answer. 
Therefore, the value of \( m \) is \( \frac{5}{8} \), corresponding to option (A).