Answer: By the end behavior of a function we mean the behavior of its values as and as

Explain in your own words the meaning of each of the followingxl lim fx 3 xl fx 5 lim xl2

(a) Can the graph of intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b) How many horizontal asymptotes can the graph of have? Sketch graphs to illustrate the possibilities

. For the function whose graph is given, state the following. (a) (b) (c) (d) (e) The equations of the asymptotes

For the function whose graph is given, state the following

510 Sketch the graph of an example of a function that satisfies all of the given conditions.5 xl lim fx 5 xl0 fx f 1 x y

510 Sketch the graph of an example of a function that satisfies all of the given conditions.xl2 lim f x xl2 lim fx xl 2 fx lim x lim f 0 0 xl lim

510 Sketch the graph of an example of a function that satisfies all of the given conditions.0, xl lim f x , xl2 xl0 lim fx xl0 f x , lim xl l

510 Sketch the graph of an example of a function that satisfies all of the given conditions.lim f xl2 lim fx xl2 lim fx xl fx 3 lim

510 Sketch the graph of an example of a function that satisfies all of the given conditions.xl0 lim f x 2, xl0 f0 3, fx 4, lim f xl2 xl 4 lim fx , xl 4 lim f x , xl fx , lim

510 Sketch the graph of an example of a function that satisfies all of the given conditions.. lim f0 0, f is even xl lim fx 2,

Guess the value of the limit by evaluating the function for and . Then use a graph of to support your guess.

. (a) Use a graph of to estimate the value of correct to two decimal places. (b) Use a table of values of to estimate the limit to four decimal places.

1314 Evaluate the limit and justify each step by indicating the appropriate properties of limits.lim xl 3x2 x  4 2x2  5x 8 fx

1314 Evaluate the limit and justify each step by indicating the appropriate properties of limits.lim xl 12x 3 5x  2 1  4x 2  3x 3 li

1538 Find the limit or show that it does not existlim xl 3x 2 2x  1 l

1538 Find the limit or show that it does not existim xl 1 x 2 x 3 x  1 li

1538 Find the limit or show that it does not existlim xl x 2 x 2  1 li

1538 Find the limit or show that it does not existlim xl 4x 3  6x 2 2 2x 3 4x  5 lim

1538 Find the limit or show that it does not existim tl st  t 2 2t t 2 l

1538 Find the limit or show that it does not existlim tl t tst 2t 32  3t 5 lim

1538 Find the limit or show that it does not existim xl 2x 2  1 2 x 1 2 x 2  x lim t

1538 Find the limit or show that it does not existlim xl x 2 sx 4  1

1538 Find the limit or show that it does not existlim xl s9x 6 x x 3  1 l

1538 Find the limit or show that it does not existlim xl s9x 6 x x 3  1 li

1538 Find the limit or show that it does not existlim 2  2x ) xl (s9x 2  x 3x)

1538 Find the limit or show that it does not existlim x l (x  sx lim 2  2x ) xl

1538 Find the limit or show that it does not existsx 2  ax sx2  bx )

1538 Find the limit or show that it does not existsx lim 2  1

1538 Find the limit or show that it does not existx 4 3x 2  x x 3 x  2 l

1538 Find the limit or show that it does not existe lim x  2 cos 3x xl

1538 Find the limit or show that it does not existlim xl x 4  x 5 lim

1538 Find the limit or show that it does not existlim xl 1  x 6 x 4  1 l

1538 Find the limit or show that it does not existlim x l arctanex li

1538 Find the limit or show that it does not existlim xl e 3x e3x e 3x  e3x lim

1538 Find the limit or show that it does not existim xl 1 ex 1  2ex li

1538 Find the limit or show that it does not existlim xl sin2 x x 2  1

1538 Find the limit or show that it does not existe2x cos x li

1538 Find the limit or show that it does not existtan1 lim ln x x

. (a) Estimate the value of by graphing the function . (b) Use a table of values of to guess the value of the limit. (c) Prove that your guess is correct.

(a) Use a graph of to estimate the value of to one decimal place. (b) Use a table of values of to estimate the limit to four decimal places. (c) Find the exact value of the limit.

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y 2x  1 x 2 f

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y x 2  1 2x 2 3x 2 y

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y 2x 2  x 1 x 2  x 2 y

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y 1  x 4 x 2 x 4 y

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y x 3 x x 2 6x  5 y

41 46 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y 2ex e x 5 y

Estimate the horizontal asymptote of the function by graphing for . Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

(a) Graph the function How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits and (b) By calculating values of , give numerical estimates of the limits in part (a). (c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]

Find a formula for a function that satisfies the following conditions:

Find a formula for a function that has vertical asymptotes and and horizontal asymptote .

A function is a ratio of quadratic functions and has a vertical asymptote and just one -intercept, . It is known that has a removable discontinuity at and . Evaluate

5256 Find the limits as and as . Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.y 2x 3 x 4 x

5256 Find the limits as and as . Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.y x 4 x6 y

5256 Find the limits as and as . Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.y x 3 x  2 2 x 1 y x

5256 Find the limits as and as . Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.y 3 x1  x 2 1 x 4 y x 3

5256 Find the limits as and as . Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.y x 2 x 2 1 2 x  2 y 3

(a) Use the Squeeze Theorem to evaluate . ; (b) Graph . How many times does the graph cross the asymptote?

By the end behavior of a function we mean the behavior of its values as and as . (a) Describe and compare the end behavior of the functions by graphing both functions in the viewing rectangles by and by . (b) Two functions are said to have the same end behavior if their ratio approaches 1 as . Show that P and Q have the same end behavior

Let and be polynomials. Find if the degree of is (a) less than the degree of and (b) greater than the degree of .

Make a rough sketch of the curve ( an integer) for the following five cases: (i) (ii) , odd (iii) , even (iv) , odd (v) , even Then use these sketches to find the following limits. (a) (b) (c) (d) f x sin xx lim xl x n lim

. Find if, for all ,10ex 21 2ex  fx  5sx sx 1 lim

(a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 Lmin. Show that the concentration of salt after minutes (in grams per liter) is (b) What happens to the concentration as ?

In Chapter 9 we will be able to show, under certain assump - tions, that the velocity of a falling raindrop at time t is where t is the acceleration due to gravity and is the terminal velocity of the raindrop. (a) Find . ; (b) Graph if and . How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

(a) By graphing and y 0.1 on a common screen, discover how large you need to make x so that . (b) Can you solve part (a) without using a graphing device? ;

Use a graph to find a number such that

. For the limit illustrate Definition 7 by finding values of that correspond to and .

. For the limit illustrate Definition 8 by finding values of that correspond to and .

For the limit illustrate Definition 9 by finding a value of that corresponds to .

(a) How large do we have to take so that ? (b) Taking in Theorem 5, we have the statement Prove this directly using Definition 7

. (a) How large do we have to take so that ? (b) Taking in Theorem 5, we have the statement Prove this directly using Definition 7.

Use Definition 8 to prove that . lim x l 1 x 0

Prove, using Definition 9, thatlim xl x 3 li

. Use Definition 9 to prove that .

Formulate a precise definition of Then use your definition to prove that

. Prove that and if these limits exist.