91Ó°ÊÓ

To evaluate the limit of a rational function that has common factors in its numerator and denominator,use the _______ _______ _______ .

When evaluating limits at infinity for complicated rational functions, you can divide the numerator and denominator by the ________ term in the denominator.

The fraction \(\frac{0}{0}\) has no meaning as a real number and therefore is called an _______ _______ .

A sequence that does not have a limit is said to ________.

GEOMETRY You create an open box from a square piece of material 24 centimeters on a side. You cut equal squares from the corners and turn up the sides. (a) Draw and label a diagram that represents the box. (b) Verify that the volume \(V\) of the box is given by \(V=4x(12-x)^2\). (C) The box has a maximum volume when \(x=4\). Use a graphing utility to complete the table and observe the behavior of the function as \(x\) approaches 4. Use the table to find \(\lim_{x \to 4} V\). (d) Use a graphing utility to graph the volume function. Verify that the volume is maximum when \(x=4\).

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{60} 7$$

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -2} \dfrac{x+2}{x^2+5x+6}$$

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -4} \dfrac{\dfrac{x}{x+2}-2}{x+4}$$

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 0} \dfrac{\sqrt{2x+1} - 1}{x}$$

In Exercises 29-36, complete the table using the function \( f(x) \), over the specified interval [a, b], to approximate the area of the region bounded by the graph of the \( y = f(x) \), the x-axis, and the vertical lines \(x=a\) and \(x=b\) and using the indicated number of rectangles. Then find the exact area as \( n \to \infty \). $$ f(x) = 20 - 2x $$ Interval \( [2, 6] \)