Limits, Continuity & Differentiability
Indeterminate Forms
Grade 12
Question:
<p>The value of <span>\(\lim_{x \to 0} \frac{e^{(1+x)^{1/x}} - e}{\tan x}\)</span> is</p>
<p>(a) <span>\(e\)</span></p>
<p>(b) <span>\(\frac{11e}{24}\)</span></p>
<p>(c) <span>\(\frac{e}{2}\)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Recognize that <span>$(1+x)^{1/x} \to e$</span> as <span>$x \to 0$</span> and use Taylor expansion for careful analysis
<p>First evaluate <span>$\lim_{x \to 0} (1+x)^{1/x} = e$</span>, so the numerator becomes <span>$e^e - e = e(e^{e-1} - 1)$</span>. Using Taylor expansion and L'Hôpital's rule on the resulting expression gives <span>$\frac{11e}{24}$</span>.</p>
Correct Answer: B