Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>If limx→∞
x2+x+1
x+1
−ax −b
= 4, then:</p>
<p>a = 1, b = 4</p>
<p>a = 1, b = −4</p>
<p>a = 2, b = −3</p>
<p>a = 2, b = 3</p>
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: Combine terms: \lim_{x \to \infty</p>} x2+x+1-(ax+b)(x+1) x+1 = \lim_{x \to \infty} (1-a)x2+(1-a-b)x+(1-b) x+1 .<p><strong>2</strong>: For the limit to be finite, the coefficient of x2 must be 0: 1 -a = 0 =\Rightarrow a = 1.</p><p><strong>3</strong>: The limit is then the coefficient ratio: 1 -a -b = 4 =\Rightarrow 1 -1 -b = 4 =\Rightarrow b = -4.</p>
Correct Answer: 2