Question:

Step 1: Evaluate the denominator at $x = -1$. The denominator is $x - 1$. Substituting $x = -1$ into the denominator yields: $$(-1) - 1 = -2$$ Since the denominator is non-zero, the limit can be found by direct substitution. Step 2: Substitute $x = -1$ into the expression. Given that $\lim_{x \to -1} \frac{x^2 - ax + b}{x - 1} = 5$, we substitute $x = -1$ into the expression: $$\frac{(-1)^2 - a(-1) + b}{(-1) - 1} = 5$$ Step 3: Simplify the expression. $$ \frac{1 + a + b}{-2} = 5 $$ Step 4: Solve for $a + b$. Multiply both sides by $-2$: $$ 1 + a + b = 5 \times (-2) $$ $$ 1 + a + b = -10 $$ Subtract $1$ from both sides: $$ a + b = -10 - 1 $$ $$ a + b = -11 $$