Question 8 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→3x4−812x2−5x−3\lim_{x \to 3} \frac{x^4 - 81}{2x^2 - 5x - 3}

Correct Answer

108/7

Exercise: EXERCISE 12.1 | Q: 8 | (Page Page 21)
For More Understanding

Explanation

This is a limit of the form 0/0 when x approaches 3. Following the method shown in examples (ii) and (iv) from the textbook context, we factorize both numerator and denominator, cancel the common factor (x-3), and then evaluate. The numerator x⁴ - 81 factors as (x-3)(x+3)(x²+9), and the denominator 2x² - 5x - 3 factors as (x-3)(2x+1). After cancellation, direct substitution gives the result.

Solution Steps

  1. Step 1: Evaluating at x = 3, numerator = 3⁴ - 81 = 81 - 81 = 0, denominator = 2(9) - 5(3) - 3 = 18 - 15 - 3 = 0. Form is 0/0.

  2. Step 2: Factorize numerator: x⁴ - 81 = (x² - 9)(x² + 9) = (x - 3)(x + 3)(x² + 9)

  3. Step 3: Factorize denominator: 2x² - 5x - 3 = 2x² - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (x - 3)(2x + 1)

  4. Step 4: Cancel common factor (x - 3) as x ≠\neq 3: lim(x→3) [(x + 3)(x² + 9)]/(2x + 1)

  5. Step 5: Substitute x = 3: = [(3 + 3)(9 + 9)]/(6 + 1) = (6×186 \times 18)/7 = 108/7

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