finding limits using properties of limits question 13, 11.2.85 points: 0 of 1 save horizontal asymptotes of…

finding limits using properties of limits question 13, 11.2.85 points: 0 of 1 save horizontal asymptotes of graphs can be described using limits. if the graph of a function f has a horizontal asymptote y = l to the right, then it can be said that lim f(x)=l. this is called a limit at infinity. note that a limit at infinity describes the end - behavior of the graph to the right. (a similar limit can be defined to describe the end - behavior to the left.) use your knowledge of horizontal asymptotes and the graphs of rational, exponential, and logistic functions to find the limit at infinity. lim 6x / (7x² + 4) as x→+∞ select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim 6x / (7x² + 4) as x→+∞ = (type an integer or a simplified fraction.) b. the limit does not exist

finding limits using properties of limits question 13, 11.2.85 points: 0 of 1 save horizontal asymptotes of graphs can be described using limits. if the graph of a function f has a horizontal asymptote y = l to the right, then it can be said that lim f(x)=l. this is called a limit at infinity. note that a limit at infinity describes the end - behavior of the graph to the right. (a similar limit can be defined to describe the end - behavior to the left.) use your knowledge of horizontal asymptotes and the graphs of rational, exponential, and logistic functions to find the limit at infinity. lim 6x / (7x² + 4) as x→+∞ select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim 6x / (7x² + 4) as x→+∞ = (type an integer or a simplified fraction.) b. the limit does not exist

Answer

Explanation:

Step1: Divide numerator and denominator by highest - power of x

Divide $\frac{6x}{7x^{2}+4}$ by $x^{2}$: $\lim_{x\rightarrow\infty}\frac{\frac{6x}{x^{2}}}{\frac{7x^{2}}{x^{2}}+\frac{4}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{\frac{6}{x}}{7 + \frac{4}{x^{2}}}$

Step2: Use limit properties

We know that $\lim_{x\rightarrow\infty}\frac{c}{x^{n}} = 0$ for $c$ constant and $n>0$. So, $\lim_{x\rightarrow\infty}\frac{6}{x}=0$ and $\lim_{x\rightarrow\infty}\frac{4}{x^{2}} = 0$. Then $\lim_{x\rightarrow\infty}\frac{\frac{6}{x}}{7+\frac{4}{x^{2}}}=\frac{\lim_{x\rightarrow\infty}\frac{6}{x}}{\lim_{x\rightarrow\infty}(7+\frac{4}{x^{2}})}=\frac{0}{7 + 0}$

Answer:

$0$