Question

The value of \( \lim_{x \to 0} \left( \frac{\tan 11x}{\tan 5x} \right) \) is:

Hint:

For small angles, \( \tan x \approx x \). This approximation is helpful for solving limits involving trigonometric functions as \( x \to 0 \).

Answer 1

Step 1: Using the small angle approximation.
For small values of \( x \), we know that \( \tan x \approx x \). Therefore, we can approximate: \[ \tan 11x \approx 11x \quad {and} \quad \tan 5x \approx 5x. \] Step 2: Substituting the approximations.
Substitute the approximations into the given expression: \[ \lim_{x \to 0} \left( \frac{\tan 11x}{\tan 5x} \right) \approx \lim_{x \to 0} \left( \frac{11x}{5x} \right). \] Step 3: Simplifying the expression.
The \( x \) terms cancel out, leaving: \[ \frac{11}{5} = 2.2. \] Thus, the value of the limit is \( \boxed{2.2} \).