Question

determine the following limit in simplest form. if the limit is infinite, state that the limit does not exist (dne).
lim_{x
ightarrowinfty}\frac{(2x - 5)(x^{3}-6)}{5x(1 + 3x^{2})}

Explanation:

Step1: Expand the numerator

$(2x - 5)(x^{3}-6)=2x\cdot x^{3}-2x\cdot6 - 5\cdot x^{3}+5\cdot6=2x^{4}-12x - 5x^{3}+30$

Step2: Expand the denominator

$5x(1 + 3x^{2})=5x+15x^{3}$

Step3: Analyze the highest - degree terms

As $x\to\infty$, the limit of a rational function is determined by the highest - degree terms of the numerator and denominator. The highest - degree term of the numerator is $2x^{4}$ and the highest - degree term of the denominator is $15x^{3}$.

Step4: Find the limit

$\lim_{x\to\infty}\frac{2x^{4}-12x - 5x^{3}+30}{5x + 15x^{3}}=\lim_{x\to\infty}\frac{2x^{4}}{15x^{3}}$ (ignoring lower - degree terms as $x\to\infty$)
$\lim_{x\to\infty}\frac{2x^{4}}{15x^{3}}=\lim_{x\to\infty}\frac{2}{15}x=\infty$

Answer:

The limit does not exist (DNE)