Question

Let \[ f(x) = \lim_{y \to 0} \frac{(1 - \cos(xy)) \tan(xy)}{y^3} \] then the number of points of intersection of \( f(x) = \sin x \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

To find the number of solutions for $f(x) = g(x)$, always check for symmetry. If both functions are odd ($f(-x)=-f(x)$), every positive root has a corresponding negative root.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We first evaluate the limit to find the functional form of \( f(x) \). The limit involves the variable \( y \) approaching zero, while \( x \) acts as a constant within the limit. We use standard trigonometric limits: \( \lim_{\theta \to 0} \frac{1-\cos \theta}{\theta^2} = \frac{1}{2} \) and \( \lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1 \).
Step 2: Key Formula or Approach:
1. Rewrite the expression to match standard limits.
2. Let \( \theta = xy \). As \( y \to 0 \), \( \theta \to 0 \).
Step 3: Detailed Explanation:
1. Evaluate \( f(x) \):
\[ f(x) = \lim_{y \to 0} \left( \frac{1 - \cos(xy)}{(xy)^2} \cdot x^2 y^2 \cdot \frac{\tan(xy)}{xy} \cdot xy \cdot \frac{1}{y^3} \right) \] \[ f(x) = \lim_{y \to 0} \left( \frac{1 - \cos(xy)}{(xy)^2} \right) \cdot \left( \frac{\tan(xy)}{xy} \right) \cdot \frac{x^3 y^3}{y^3} \] \[ f(x) = \frac{1}{2} \cdot 1 \cdot x^3 = \frac{x^3}{2} \] 2. Find intersection points of \( \frac{x^3}{2} = \sin x \).
3. By analyzing the graphs of \( y = \frac{x^3}{2} \) (a cubic curve) and \( y = \sin x \) (a wave):
- They intersect at \( x = 0 \) (since \( 0 = \sin 0 \)).
- For \( x > 0 \), \( \frac{x^3}{2} \) grows faster than \( \sin x \), providing 1 intersection.
- Due to odd symmetry (\( f(-x) = -f(x) \)), there is 1 intersection for \( x < 0 \).
Step 4: Final Answer:
The number of points of intersection is 3 (\( x=0 \) and two non-zero roots).