Exam 1: Functions and Limits

Find the interval(s) where is continuous.

Let and . Find .

Find a function g that agrees with f for and is continuous on .

Find if

If , then if exists, to what value does it converge?

The symbol can be used to denote the greatest integer function, which is defined by the greatest integer n such that Use the graph of the function to find the indicated limit, if it exists.

The following figure shows a portion of the graph of a function f defined on the interval . Sketch the complete graph of f if it is known f is odd.

Find the vertical asymptotes of the function.

You are given that , , and . Find the limit .

Find the numbers, if any, where the function is discontinuous.

Find the numbers, if any, where the function is discontinuous.

Use the precise definition of a limit to prove that

The graph of the function follows. Choose the graph of

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A measured in of the caves by the equation (a) The cave called mission impossible near puebla, mexico, has suface area of . How many species of bats would expect to find in that cave? (b) If you discover that species of bats live in cave estimate the area of the cave.

Refer to the graph of the function f in the following figure. a.Find . b.Find the value of x for which (i) and (ii) . c.Find the domain and range of f.

Find a, such that the function has the domain .

Complete the table by computing at the given values of x, accurate to five decimal places. Use the results to guess at the indicated limit, if it exists, to three decimal places.

Find the limit.

Estimate the value of the limit by graphing the function . State your answer correct to two decimal places.

Find the function such that . and ,