Question 29 of 70intermediate🔧 ApplyShort Answer2 marks

If the function f(x)f(x) satisfies lim⁡x→1f(x)−2x2−1=π\lim_{x \to 1} \frac{f(x) - 2}{x^2 - 1} = \pi, evaluate lim⁡x→1f(x)\lim_{x \to 1} f(x).

Correct Answer

2

Exercise: EXERCISE 12.1 | Q: 31 | (Page Page 22)
For More Understanding

Explanation

This question tests the understanding of when a limit exists as a finite value. From the context, we see examples where limits are evaluated and the concept that limits must be finite values. When the denominator approaches 0 and the limit exists as a finite non-zero value (π), the numerator must also approach 0. This is because if the numerator approached a non-zero value while the denominator approached 0, the limit would be undefined (infinite). The context shows that limits follow algebraic properties like lim[g(x) + h(x)] = lim g(x) + lim h(x).

Solution Steps

  1. Step 1: Given that lim(x→1) [f(x) - 2]/(x² - 1) = π, which is a finite value.

  2. Step 2: As x → 1, the denominator (x² - 1) → 1² - 1 = 0.

  3. Step 3: For the limit to exist as a finite value π, the numerator must also approach 0 (otherwise the limit would be undefined).

  4. Step 4: Therefore, lim(x→1) [f(x) - 2] = 0.

  5. Step 5: Using limit properties: lim(x→1) f(x) - lim(x→1) 2 = 0, which gives lim(x→1) f(x) - 2 = 0.

  6. Step 6: Hence, lim(x→1) f(x) = 2.

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