Calcular los límites cuando \( x \) tiende a cero de las funciones: (a) \( \frac{\cos x-\frac{x^{2}}{2}-1}{x^{3}} \) (b) \( \frac{\sin x+e^{x}-1}{x} \) (c) \( \frac{e^{x}-1}{x} \)

A hotdog vendor must pay a monthly fee to operate a food cart in the city park. The cost in dollars of selling \( x \) hundred hotdogs in a month can be approximated by \( \mathrm{C}(\mathrm{x})=150+60 \mathrm{x} \). Complete parts (a) through (d) below. (a) Find the average-cost function. The average-cost function is \( \overline{\mathrm{C}}(\mathrm{x})=\square \) (b) Find and interpret the average cost at sales levels of 1000 hotdogs and 5500 hotdogs. The average cost at sales levels of 1000 hotdogs is \( \$ \square / 100 \) hotdogs. (Round to the nearest cent as needed.) The average cost at sales levels of 5500 hotdogs is \( \$ \square / 100 \) hotdogs. (Round to the nearest cent as needed.) Interpret these results, choose the correct answer below. Take the sales level of 1000 hotdogs as an example. Select the correct choice below and fill in the answer box to complete your choice. A. At a sales level of 1000 hotdogs, the average cost is \( \$ \square \) per 100 hotdogs. B. At a sales level of 1000 hotdogs, if an additional 100 hotdogs are sold, the average cost increases by approximately \( \$ \square \) per 100 hotdogs.

A car's velocity is given by the function \( v(t) = 5t^2 + 3 \). Calculate the distance traveled by the car from time \( t = 0 \) to \( t = 4 \).

\( \int e ^ { \frac { - 1 } { x } } d x \)

The revenue (in thousands of dollars) from producing \( x \) units of an item is modeled by \( R(x)=13 x-0.01 x^{2} \). a. Find the average rate of change in revenue as \( x \) changes from 1004 to 1005 . b. Find the marginal revenue at \( x=500 \).

(c) Use Bisection method to solve \( x^{3}-9 x+1=0 \) for the root between \( x=2 \) and \( x=4 \). 5 marks) (d) Using \( \sin (0.1)=0.09983 \) and \( \sin (0.2)=0.198 \), find an appropriate value of \( \sin (0.15) \) by using Langranges interpolation.