Question

$\lim\limits_{x \to \infty} \frac{(6 - x)(3 + 10x)}{(3 - 8x)(7 + 10x)}$
options: -6/7, 6/7, -1/8, 1/8
question 3
$\lim\limits_{x \to -\infty} \frac{3}{e^x + 9}$
options: 0, $-\infty$, 3, 1/3

Explanation:

First Limit Problem: $\boldsymbol{\lim_{x \to \infty} \frac{(6 - x)(3 + 10x)}{(3 - 8x)(7 + 10x)}}$

Step1: Expand numerator and denominator

Numerator: $(6 - x)(3 + 10x) = 18 + 60x - 3x - 10x^2 = -10x^2 + 57x + 18$
Denominator: $(3 - 8x)(7 + 10x) = 21 + 30x - 56x - 80x^2 = -80x^2 - 26x + 21$

Step2: Divide by highest power of $x$

Answer:

Step1: Expand numerator and denominator

Numerator: $(6 - x)(3 + 10x) = 18 + 60x - 3x - 10x^2 = -10x^2 + 57x + 18$
Denominator: $(3 - 8x)(7 + 10x) = 21 + 30x - 56x - 80x^2 = -80x^2 - 26x + 21$

Step2: Divide by highest power of $x$