use lhopitals rule to find the limit. note that in this problem, neither algebraic simplification nor the…

use lhopitals rule to find the limit. note that in this problem, neither algebraic simplification nor the theorem for limits of rational functions at infinity provides an alternative to lhopitals rule. lim(x→∞) x / ln x^5 select the correct choice below and, if necessary, fill in the answer box to complete your choice. oa. lim(x→∞) x / ln x^5 = (simplify your answer.) ob. the limit does not exist.

use lhopitals rule to find the limit. note that in this problem, neither algebraic simplification nor the theorem for limits of rational functions at infinity provides an alternative to lhopitals rule. lim(x→∞) x / ln x^5 select the correct choice below and, if necessary, fill in the answer box to complete your choice. oa. lim(x→∞) x / ln x^5 = (simplify your answer.) ob. the limit does not exist.

Answer

Explanation:

Step1: Check indeterminate form

As $x\to\infty$, we have $\lim_{x\to\infty}\frac{x}{\ln x^{5}}=\frac{\infty}{\infty}$, so L'Hopital's rule can be applied.

Step2: Differentiate numerator and denominator

The derivative of the numerator $y = x$ is $y'=1$. The derivative of the denominator $y=\ln x^{5}=5\ln x$, and its derivative is $y'=\frac{5}{x}$ according to the chain - rule and the derivative formula of $\ln u$.

Step3: Find the new limit

By L'Hopital's rule, $\lim_{x\to\infty}\frac{x}{\ln x^{5}}=\lim_{x\to\infty}\frac{1}{\frac{5}{x}}=\lim_{x\to\infty}\frac{x}{5}$.

Step4: Evaluate the limit

As $x\to\infty$, $\lim_{x\to\infty}\frac{x}{5}=\infty$. But we made a mistake above, we should first simplify $\ln x^{5}=5\ln x$. Now, applying L'Hopital's rule to $\lim_{x\to\infty}\frac{x}{5\ln x}$ (which is in $\frac{\infty}{\infty}$ form). The derivative of $x$ is $1$ and the derivative of $5\ln x$ is $\frac{5}{x}$. So $\lim_{x\to\infty}\frac{x}{5\ln x}=\lim_{x\to\infty}\frac{1}{\frac{5}{x}}=\lim_{x\to\infty}\frac{x}{5}=\infty$.

Answer:

B. The limit does not exist.