https://file.aiquickdraw.com/mathbot/file/1725747243985f0foukrh.jpg

The limit presented in the image is:

\[
\lim_{x \to \pi/2} \frac{2 \cos x}{\cos x}
\]

To solve this, we can simplify the expression inside the limit. Since the cosine terms are the same in the numerator and denominator, they can cancel each other out, except at points where \(\cos x = 0\), which does not include \(x = \pi/2\). The simplified expression is:

\[
2 \cdot \frac{\cos x}{\cos x} = 2
\]

As \(x \to \pi/2\), \(\cos x \to 0\), but the division has been resolved before the limit, so:

\[
\lim_{x \to \pi/2} 2 = 2
\]

Thus, the limit is \(2\).

Would you like more details on this or do you have any further questions? Here are some related questions you might consider:

1. What happens to the limit if the limit point changes to \(x \to 0\)?
2. How would the limit change if the expression was \( \frac{2 \sin x}{\cos x} \)?
3. Can you explain the behavior of \( \cos x \) and \( \sin x \) as \( x \) approaches \( \pi/2 \)?
4. What is the general approach to solving limits involving trigonometric functions?
5. How would adding a term to the numerator, like \( + \sin x \), affect the limit?

**Tip:** When dealing with limits involving trigonometric functions, always consider the periodic properties and the specific values of the functions at important points like \( \pi/2, 0, \pi, \) and \( 2\pi \).

Find the limit of the trigonometric function: lim_{x→π/2} (2cos(x))/cos(x).

Learn how to find the limit of the trigonometric function lim_{x→π/2} (2cos(x))/cos(x) with a detailed step-by-step solution. This problem covers key calculus concepts, including trigonometric limits and the simplification of rational expressions. Suitable for high school students in Grades 11-12.

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