Question 6 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→0(x+1)5−1x\lim_{x \to 0} \frac{(x + 1)^5 - 1}{x}

Correct Answer

5

Exercise: EXERCISE 12.1 | Q: 6 | (Page Page 21)
For More Understanding

Explanation

The problem is solved by substituting y = x+1, which transforms the limit into the standard form lim_{y→1} (y5y^{5} - 1)/(y-1). The textbook provides the theorem lim_{x→a} (xnx^{n} - ana^{n})/(x - a) = n an−1a^{n-1}, which directly gives the result 5.

Solution Steps

  1. Step 1: Let y = x + 1. Then as x → 0, y → 1.

  2. Step 2: Rewrite the limit: lim_{x→0} ((x+1)^5 - 1)/x = lim_{y→1} (y5y^{5} - 1)/(y-1).

  3. Step 3: Apply the theorem: lim_{y→1} (y5y^{5} - 1)/(y-1) = 5 * 15−11^{5-1} = 5.

  4. Step 4: Hence, the required limit is 5.

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