Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>If limx→0[1 + x ln(1 + b2)]1/x = 2b sin2 θ, b > 0 and θ ∈(−π, π], then the value of θ is:</p>
\pm \pi/4
\pm \pi/3
\pm \pi/6
\pm \pi/2
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: This is a 1\inftyform. Limit = elimx\to 0 1</p> x[x ln(1+b2)] = eln(1+b2) = 1 + b2.<p><strong>2</strong>: Given 1 + b2 = 2b sin2 \theta =\Rightarrow sin2 \theta = b2+1</p> 2b .<p><strong>3</strong>: Since b2+1</p> 2b \geq1 (by AM-GM) and sin2 \theta \leq1, it must be that sin2 \theta = 1.<p><strong>4</strong>: This occurs when \theta = \pm\pi/2.</p>
Correct Answer: 4