91Ó°ÊÓ

Evaluate the limit, if it exists. $$ \lim _{x \rightarrow+\infty} \frac{(\ln x)^{3}}{x} $$

Find the Taylor polynomial of degree \(n\) with the Lagrange form of the remainder at the number \(a\) for the function defined by the given equation. $$ f(x)=(1+x)^{3 / 2} ; a=0 ; n=3 $$

Determine whether the improper integral is convergent or divergent. If it is convergent, evaluate it. $$ \int_{e}^{+\infty} \frac{d x}{x(\ln x)^{2}} $$

Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\tanh 2 x}{\tanh x} $$

The continuous flow of profit for a company is increasing with time, and at \(t\) years the number of dollars in the profit per year is proportional to \(t\). Show that the present value of the company is inversely proportional to \(t^{2}\), where \(100 i\) percent is the interest rate compounded continuously.