Question 13 of 14intermediate🔧 ApplyNumerical2 marks

Evaluate the following limit, if exist: lim⁡x→0log⁡e(1+2x)x\lim_{x \to 0} \frac{\log_e (1 + 2x)}{x}

Correct Answer

2

Exercise: Exercise 13.2 | Q: 7 | (Page Page 6)
For More Understanding

Explanation

This question tests the application of the standard logarithmic limit result. The student needs to recognize that the given limit can be transformed using substitution to match the standard form proven in Theorem 7 of the textbook. The key insight is to substitute y = 2x, which allows the expression to be rewritten in terms of the known limit.

Solution Steps

  1. Step 1: Let y = 2x. Then as x → 0, we have y → 0.

  2. Step 2: Rewrite the limit: lim(x→0) [log_e(1 + 2x)/x] = lim(y→0) [log_e(1 + y)/(y/2)]

  3. Step 3: Simplify: = lim(y→0) [log_e(1 + y)/y] × 2

  4. Step 4: Apply the standard result lim(x→0) [log_e(1 + x)/x] = 1 (from Theorem 7)

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

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