Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>The limit limx→0
3√
1+x2−4√1−2x
x+x2
is equal to:</p>
Step-by-Step Solution
Key Concept: Use Binomial Expansion for small x: (1 + x)n \approx1 + nx for x ≪1.
<p><strong>1</strong>: </p> 3\sqrt 1 + x2 = (1 + x2)1/3 \approx1 + x2/3.<p><strong>2</strong>: </p> 4\sqrt{1} -2x = (1 -2x)1/4 \approx1 + (1/4)(-2x) = 1 -x/2.<p><strong>3</strong>: Numerator = (1 + x2/3) -(1 -x/2) = x/2 + x2/3.</p><p><strong>4</strong>: Denominator = x + x2.</p><p><strong>5</strong>: Limit = \lim_{x \to 0</p>} x/2+x2/3 x+x2 = \lim_{x \to 0} 1/2+x/3 1+x = 1/2.
Correct Answer: 2