Question

If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]

• A. 0
• B. 5
• C. -1
• D. -5
• E. 1

Hint:

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.

Answer 1

Answer: (E)

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 \( (x - 1) \times (\text{some expression}) \), 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).