https://file.aiquickdraw.com/mathbot/file/1726564647381dbve26ez.jpg

The problem shown in the image asks for the limit:

\[
\lim_{{x \to \frac{\pi}{3}}} \frac{2 \tan x - \sin x}{\cos x}
\]

### Step-by-Step Solution:

1. **Evaluate the trigonometric functions at \(x = \frac{\pi}{3}\):**
   - \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\)
   - \(\cos \frac{\pi}{3} = \frac{1}{2}\)
   - \(\tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\)

2. **Substitute these values into the expression:**
   \[
   \frac{2 \tan \frac{\pi}{3} - \sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{2 \cdot \sqrt{3} - \frac{\sqrt{3}}{2}}{\frac{1}{2}}
   \]

3. **Simplify the numerator:**
   \[
   2 \cdot \sqrt{3} - \frac{\sqrt{3}}{2} = \frac{4 \sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{2}
   \]

4. **Divide by \(\cos \frac{\pi}{3}\):**
   \[
   \frac{\frac{3 \sqrt{3}}{2}}{\frac{1}{2}} = 3 \sqrt{3}
   \]

So, the limit is:

\[
\lim_{{x \to \frac{\pi}{3}}} \frac{2 \tan x - \sin x}{\cos x} = 3 \sqrt{3}
\]

Would you like more details or have any further questions?

Here are five related questions:
1. How do you find the tangent of an angle using sine and cosine?
2. What are the key steps in calculating limits involving trigonometric functions?
3. How can L'Hopital's Rule be used in cases where limits lead to indeterminate forms?
4. How do trigonometric identities help in simplifying complex expressions?
5. What are the common trigonometric values for angles like \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \)?

**Tip:** Always evaluate known trigonometric values for standard angles when calculating limits involving trigonometric expressions.

Solve the limit lim_(x→π/3) (2tanx - sinx) / cosx step by step. Learn the use of trigonometric identities and limit properties to find the solution 3√3. Ideal for Grade 12 students studying calculus.

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