Kerala KEAM 5 June 2024 Solved Paper

We are given the limit: $$\lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5$$Step 1: First, factor the numerator. The expression $x^2 - ax - b$ should be factorable in the form $(x - 1)(some expression)$, because the denominator is $x - 1$.We need to ensure that the numerator has a factor of $(x - 1)$, so substitute $x = 1$ into the numerator:$$x^2 - ax - b = 1^2 - a(1) - b = 1 - a - b$$For the expression to have a factor of $(x - 1)$, this must be zero. Therefore: $$1 - a - b = 0$$ $$a + b = 1$$Thus, $a + b = 1$.Therefore, the correct answer is option (E).Quick Tip: When dealing with limits involving rational functions, ensure that the numerator has a factor that cancels with the denominator. Substitute the limiting value into the numerator to find the necessary conditions for cancellation.