Question 7 of 14beginner🔧 ApplyNumerical2 marks

Evaluate the following limit, if exist: lim⁡x→0e4x−1x\lim_{x \to 0} \frac{e^{4x} - 1}{x}

Correct Answer

4

Exercise: Exercise 13.2 | Q: 1 | (Page Page 6)
For More Understanding

Explanation

This limit is evaluated using the standard result proved in Theorem 6 of the textbook, which states that lim(x→0) (exe^{x} - 1)/x = 1. The solution follows the same method as Example 5, where the coefficient of x in the exponent becomes the final answer after applying the standard limit.

Solution Steps

  1. Step 1: We have lim(x→0) (e^(4x) - 1)/x

  2. Step 2: Multiply and divide by 4: lim(x→0) [(e^(4x) - 1)/(4x)] × 4

  3. Step 3: Let y = 4x. As x → 0, y → 0. Therefore, = 4 × lim(y→0) (eye^{y} - 1)/y

  4. Step 4: Using the standard result lim(x→0) (exe^{x} - 1)/x = 1, we get = 4×14 \times 1 = 4

PreviousNext