Question 12 of 14intermediate🔧 ApplyNumerical3 marks

Evaluate the following limit, if exist: lim⁡x→0x(ex−1)1−cos⁡x\lim_{x \to 0} \frac{x(e^x - 1)}{1 - \cos x}

Correct Answer

2

Exercise: Exercise 13.2 | Q: 6 | (Page Page 6)
For More Understanding

Explanation

This question tests the application of standard limits. The student needs to recognize that the limit involves two standard results: lim(x→0) (exe^{x} - 1)/x = 1 (Theorem 6) and lim(x→0) sin x / x = 1 (from Example 6). The key insight is to use the trigonometric identity 1 - cos x = 2sin²(x/2) to transform the denominator into a form where the standard limit for sin x/x can be applied.

Solution Steps

  1. Step 1: Rewrite the denominator using the identity 1 - cos x = 2sin²(x/2)

    lim(x→0) [x(exe^{x} - 1)]/(1 - cos x) = lim(x→0) [x(exe^{x} - 1)]/[2sin²(x/2)]

  2. Step 2: Rearrange the expression to separate standard limits

    = lim(x→0) [(exe^{x} - 1)/x] × [x²/(2sin²(x/2))]

  3. Step 3: Evaluate the first part using Theorem 6

    lim(x→0) (exe^{x} - 1)/x = 1

  4. Step 4: Evaluate the second part by substituting t = x/2

    As x → 0, t → 0

    x²/(2sin²(x/2)) = (2t)²/(2sin²t) = 4t²/(2sin²t) = 2(t/sin t)²

    Since lim(t→0) sin t/t = 1, we have lim(t→0) t/sin t = 1

    Therefore, lim(x→0) x²/(2sin²(x/2)) = 2×12 \times 1 = 2

  5. Step 5: Combine the results

    lim(x→0) [x(exe^{x} - 1)]/(1 - cos x) = 1×21 \times 2 = 2

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