Assume that lim x f (x) = L and limxL g(x) = Which of the following statements are

Assume that and Which of the following statements are correct? (a) x = L is a vertical asymptote of g. (b) y = L is a horizontal asymptote of g. (c) x = L is a vertical asymptote of f. (d) y = L is a horizontal asymptote of f.

What are the following limits? (a) lim x x3 (b) lim x x3 (c) lim x x4

Sketch the graph of a function that approaches a limit as x but does not approach a limit (either finite or infinite) as x .

What is the sign of a if f (x) = ax3 + x + 1 satisfies lim x f (x) = ?

What is the sign of the coefficient multiplying x7 if f is a polynomial of degree 7 such that lim x f (x) = ?

Explain why lim x sin 1 x exists but limx0 sin 1 x does not exist. What is lim x sin 1 x ?

In Exercises 716, evaluate the limit. lim x x x + 9

In Exercises 716, evaluate the limit. lim x 3x2 + 20x 4x2 + 9

In Exercises 716, evaluate the limit. lim x 3x2 + 20x 2x4 + 3x3 29

In Exercises 716, evaluate the limit. lim x 4 x + 5

In Exercises 716, evaluate the limit. lim x 7x 9 4x + 3

In Exercises 716, evaluate the limit. lim x 9x2 2 6 29x

In Exercises 716, evaluate the limit. lim x 7x2 9 4x + 3

In Exercises 716, evaluate the limit. lim x 5x 9 4x3 + 2x + 7

In Exercises 716, evaluate the limit. lim x 3x3 10 x + 4

In Exercises 716, evaluate the limit. lim x 2x5 + 3x4 31x 8x4 31x2 + 12

In Exercises 1722, find the horizontal asymptotes. f (x) = 2x2 3x 8x2 + 8

In Exercises 1722, find the horizontal asymptotes. f (x) = 8x3 x2 7 + 11x 4x4

In Exercises 1722, find the horizontal asymptotes. f (x) = 36x2 + 7 9x + 4

In Exercises 1722, find the horizontal asymptotes. f (x) = 36x4 + 7 9x2 + 4

In Exercises 1722, find the horizontal asymptotes. f (t) = et 1 + et

In Exercises 1722, find the horizontal asymptotes. f (t) = t1/3 (64t2 + 9)1/6

In Exercises 2330, evaluate the limit. lim x 9x4 + 3x + 2 4x3 + 1

In Exercises 2330, evaluate the limit. lim x x3 + 20x 10x 2

In Exercises 2330, evaluate the limit. x 8x2 + 7x1/3 16x4 + 6

In Exercises 2330, evaluate the limit. lim x 4x 3 25x2 + 4x

In Exercises 2330, evaluate the limit. lim t t4/3 + t1/3 (4t2/3 + 1)2

In Exercises 2330, evaluate the limit. lim t t4/3 9t1/3 (8t4 + 2)1/3

In Exercises 2330, evaluate the limit. lim x |x| + x x + 1

In Exercises 2330, evaluate the limit. lim t 4 + 6e2t 5 9e3t

Determine lim x tan1 x. Explain geometrically.

Show that lim x( x2 + 1 x) = 0. Hint: Observe that x2 + 1 x = 1 x2 + 1 + x

According to the MichaelisMenten equation (Figure 7), when an enzyme is combined with a substrate of concentration s (in millimolars), the reaction rate (in micromolars/min) is R(s) = As K + s (A, K constants) (a) Show, by computing lim s R(s), that A is the limiting reaction rate as the concentration s approaches . (b) Show that the reaction rate R(s) attains one-half of the limiting value A when s = K. (c) For a certain reaction, K = 1.25 mM and A = 0.1. For which concentration s is R(s) equal to 75% of its limiting value?

Suppose that the average temperature of Earth is T (t) = 283 + 3(1 e0.03t) kelvins, where t is the number of years since 2000.(a) Calculate the long-term average L = lim t T (t). (b) At what time is T (t) within one-half a degree of its limiting value?

In Exercises 3542, calculate the limit.. lim x( 4x4 + 9x 2x2)

In Exercises 3542, calculate the limit. lim x( 9x3 + x x3/2)

In Exercises 3542, calculate the limit. lim x(2 x x + 2)

In Exercises 3542, calculate the limit. lim x 1 x 1 x + 2

In Exercises 3542, calculate the limit. lim x (ln(3x + 1) ln(2x + 1))

In Exercises 3542, calculate the limit. lim x  ln( 5x2 + 2) ln x

In Exercises 3542, calculate the limit. lim x tan1  x2 + 9 9 x

In Exercises 3542, calculate the limit. lim x tan1 1 + x 1 x

Let P (n) be the perimeter of an n-gon inscribed in a unit circle (Figure 8). (a) Explain, intuitively, why P (n) approaches 2 as n . (b) Show that P (n) = 2n sin  n  . (c) Combine (a) and (b) to conclude that lim n n sin  n  = 1. (d) Use this to give another argument that lim0 sin = 1. n = 6 n = 9 n = 12

Physicists have observed that Einsteins theory of special relativity reduces to Newtonian mechanics in the limit as c , where c is the speed of light. This is illustrated by a stone tossed up vertically from ground level so that it returns to Earth 1 s later. Using Newtons Laws, we find that the stones maximum height is h = g/8 meters (g = 9.8 m/s2). According to special relativity, the stones mass depends on its velocity divided by c, and the maximum height is h(c) = c c2/g2 + 1/4 c2/g Prove that lim c h(c) = g/8.

Every limit as x can be rewritten as a one-sided limit as t 0+, where t = x1. Setting g(t) = f (t1), we have lim x f (x) = lim t0+ g(t) Show that lim x 3x2 x 2x2 + 5 = lim t0+ 3 t 2 + 5t2 , and evaluate using the Quotient Law.

Rewrite the following as one-sided limits as in Exercise 45 and evaluate. (a) lim x 3 12x3 4x3 + 3x + 1 (b) lim x e1/x (c) lim x x sin 1 x (d) lim x ln x + 1 x 1

Let G(b) = lim x(1 + bx ) 1/x for b 0. Investigate G(b) numerically and graphically for b = 0.2, 0.8, 2, 3, 5 (and additional values if necessary). Then make a conjecture for the value of G(b) as a function of b. Draw a graph of y = G(b). Does G appear to be continuous? We will evaluate G(b) using LHpitals Rule in Section 4.5 (see Exercise 69 there).