Question

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

Hint:

For any limit of the form \( \lim_{x \to 0} \frac{\sin ax}{bx} \), the result is simply the ratio of the coefficients, \( \frac{a}{b} \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A function \( f(x) \) is continuous at \( x = a \) if the limit of the function as \( x \) approaches \( a \) is equal to the value of the function at \( a \).
\[ \lim_{x \to 0} f(x) = f(0) \] Step 2: Key Formula or Approach:
Use the standard trigonometric limit: \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).
Step 3: Detailed Explanation:
Evaluate the limit:
\[ \lim_{x \to 0} \frac{\sin 8x}{5x} = \frac{1}{5} \lim_{x \to 0} \frac{\sin 8x}{x} \] Multiply and divide by 8 to match the angle:
\[ = \frac{8}{5} \lim_{x \to 0} \frac{\sin 8x}{8x} \] Using the standard limit, \( \lim_{8x \to 0} \frac{\sin 8x}{8x} = 1 \):
\[ = \frac{8}{5} \times 1 = \frac{8}{5} \] Given that the function is continuous, \( f(0) = \frac{8}{5} \).
From the definition: \( f(0) = m + 1 \).
Equating both:
\[ m + 1 = \frac{8}{5} \] \[ m = \frac{8}{5} - 1 = \frac{8 - 5}{5} = \frac{3}{5} \] Step 4: Final Answer:
The value of \( m \) is \( \frac{3}{5} \).