Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let f(x) = tan x
x , then the value of limx→0([f(x)] + x2)1/{f(x)} is:</p>
<p>e−3</p>
<p>e3</p>
<p>e2</p>
<p>non-existent</p>
Step-by-Step Solution
Key Concept: General
<p>As x \to 0, f(x) \to 1+. Thus [f(x)] = 1. {f(x)} = f(x) -1 = tan x-x</p> x . Limit = elimx\to 0 x2 {f(x)} = elim x3 tan x-x . Using expansion tan x \approxx + x3/3, limit = e lim x3 x3/3 = e3.
Correct Answer: 2