Question 20 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→π2tan⁡2xx−π2\lim_{x \to \frac{\pi}{2}} \frac{\tan 2x}{x - \frac{\pi}{2}}

Correct Answer

2

Exercise: EXERCISE 12.1 | Q: 22 | (Page Page 22)
For More Understanding

Explanation

This is a standard limit problem involving trigonometric functions. The key is to use substitution to transform the limit into a form where we can apply the standard result that lim(x→0) tan(x)/x = 1, which is provided in the textbook context. By substituting t = x - π/2, we convert the limit to evaluate at t→0, making the standard formula applicable.

Solution Steps

  1. Step 1: Let t = x - π/2. As x → π/2, we have t → 0.

  2. Step 2: Rewrite tan(2x) = tan(2(t + π/2)) = tan(2t + π) = tan(2t) [since tan has period π]

  3. Step 3: The limit becomes lim(t→0) tan(2t)/t = lim(t→0) 2·tan(2t)/(2t)

  4. Step 4: Using the standard result lim(x→0) tan(x)/x = 1, we get lim(t→0) tan(2t)/(2t) = 1

  5. Step 5: Therefore, the limit = 2×12 \times 1 = 2

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