Limits, Continuity & Differentiability
Continuity
Grade 12
Question:
<p>Let <em>f</em> be continuous on <strong>R</strong>. If <em>f</em>(0) = 1 and <em>f</em> is defined as <em>f</em>(1/4<sup><em>n</em></sup>) involves <em>sin e<sup>n</sup></em> and exponential terms, and <em>f</em> is continuous on <strong>R</strong>, find <em>f</em>(0).</p>
Step-by-Step Solution
Key Concept: When a function is defined piecewise with a limit-based formula at specific points, continuity at 0 forces the limit of the sequence of function values to equal f(0). Use the squeeze theorem or direct limit evaluation combined with the continuity condition.
<p><strong>Step 1:</strong> Recognize that f is continuous on ℝ, meaning f is continuous at x = 0.</p><p><strong>Step 2:</strong> By definition of continuity at x = 0: lim(x→0) f(x) = f(0).</p><p><strong>Step 3:</strong> We are given that f(0) = 1.</p><p><strong>Step 4:</strong> Since the sequence {1/4n} → 0 as n → ∞, and f is continuous at 0, we have lim(n→∞) f(1/4n) = f(0) = 1.</p><p><strong>Step 5:</strong> The continuity condition confirms that regardless of the specific form of f at the points 1/4n (involving sin and exponential terms), the limiting behavior is dictated by f(0).</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1