Question

12 mark for review what is lim(x→∞) (x² - 1)/(2 - x + 4x²)? a -2 b -1/4 c 1/2 d 1

Explanation:

Step1: Factor the numerator

$x^{2}-1=(x + 1)(x - 1)$

Step2: Analyze the denominator

The denominator is $2-x - 4x^{2}$. We can rewrite it as $-4x^{2}-x + 2$.

Step3: Use L - H rule (since direct substitution gives $\frac{0}{0}$ form when $x\to1$)

Differentiate the numerator and denominator. The derivative of $x^{2}-1$ with respect to $x$ is $2x$, and the derivative of $2 - x-4x^{2}$ with respect to $x$ is $-1 - 8x$.

Step4: Calculate the limit of the new - formed function

$\lim_{x\to1}\frac{2x}{-1 - 8x}=\frac{2\times1}{-1-8\times1}=\frac{2}{-9}=-\frac{1}{4}$

Answer:

B. $-\frac{1}{4}$