Question 7 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→23x2−x−10x2−4\lim_{x \to 2} \frac{3x^2 - x - 10}{x^2 - 4}

Correct Answer

114\frac{11}{4}

Exercise: EXERCISE 12.1 | Q: 7 | (Page Page 21)
For More Understanding

Explanation

This question involves evaluating a limit that results in an indeterminate form (0/0). Following the pattern shown in the textbook context (examples ii, iii, iv), when direct substitution gives 0/0 form, we factorize both numerator and denominator, cancel common factors, and then evaluate. The numerator 3x² - x - 10 factorizes to (x-2)(3x+5) and denominator x² - 4 factorizes to (x-2)(x+2). After canceling (x-2), we substitute x=2 to get the final answer.

Solution Steps

  1. Step 1: Evaluating at x = 2, we get numerator = 3(2)² - 2 - 10 = 12 - 2 - 10 = 0 and denominator = (2)² - 4 = 4 - 4 = 0. Hence, it is of the form 0/0.

  2. Step 2: Factorizing numerator: 3x² - x - 10 = (x - 2)(3x + 5). Factorizing denominator: x² - 4 = (x - 2)(x + 2).

  3. Step 3: Hence, lim(x→2) [(3x² - x - 10)/(x² - 4)] = lim(x→2) [(x - 2)(3x + 5)/((x - 2)(x + 2))] = lim(x→2) [(3x + 5)/(x + 2)] as x ≠\neq 2.

  4. Step 4: Substituting x = 2: = (3(2) + 5)/(2 + 2) = (6 + 5)/4 = 11/4.

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