Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>limx→∞
11/x+21/x+31/x+···+n1/x
n
nx
where n ∈N is equal to:</p>
<p>n!</p>
<p>1</p>
<p>1
n!</p>
<p>0</p>
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: 1\inftyform. Limit = e</p> \lim_{x \to \inftynx} Pn k=1 k1/x n -1 = elimx\to \inftyx Pn k=1(k1/x-1).<p><strong>2</strong>: Using \lim_{t \to 0} at-1</p> t = ln a, let t = 1/x:<p><strong>3</strong>: Exponent = Pn</p> k=1 \lim_{t \to 0} kt-1 t = Pn k=1 ln k = ln(n!).<p><strong>4</strong>: Result = eln(n!) = n!.</p>
Correct Answer: 1