Limits, Continuity & Differentiability
Continuity at a Point
Grade 12
Question:
<p>If the function <span style='display:inline-block'>f(x) = {a|p - x| + 1, x ≤ 5; b|x - p| + 3, x > 5}</span> is continuous at x = 5, then the value of a - b is</p>
<p>(a) <span style='display:inline-block'>\frac{p+5}{p+5}\)</span></p>
<p>(b) <span style='display:inline-block'>\frac{p-5}{5-p}\)</span></p>
<p>(c) <span style='display:inline-block'>\frac{2}{5}\)</span></p>
<p>(d) <span style='display:inline-block'>\frac{2}{3}\)</span></p>
Step-by-Step Solution
Key Concept: For a function to be continuous at a point, the left-hand limit, right-hand limit, and function value must all be equal.
<p>For continuity at x = 5, left limit = right limit = f(5). Apply continuity condition at x = 5 to get the relationship between a and b, then solve for a - b.</p>
Correct Answer: A