Question 14 of 70beginner🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→0cos⁡xπ−x\lim_{x \to 0} \frac{\cos x}{\pi - x}

Correct Answer

1π\frac{1}{\pi}

Exercise: EXERCISE 12.1 | Q: 16 | (Page Page 22)
For More Understanding

Explanation

This question tests the direct substitution property of limits. From the context, we see in Illustration 7 that lim(x→0) cos x = 1. The denominator π - x is a polynomial function, and limits of polynomials can be evaluated by direct substitution. Since the denominator approaches π (which is non-zero), we can apply the quotient rule for limits.

Solution Steps

  1. Step 1: Identify that as x → 0, we need to evaluate both numerator and denominator separately.

  2. Step 2: From the context (Illustration 7), we know that lim(x→0) cos x = 1.

  3. Step 3: For the denominator: lim(x→0) (π - x) = π - 0 = π.

  4. Step 4: Apply the quotient rule for limits: lim(x→0) [cos x/(π - x)] = [lim(x→0) cos x] / [lim(x→0) (π - x)] = 1/π.

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