Question: Find the limit or show that it
Find the limit or show that it does not exist. 
Transcribed Image Text:
sin?x 2. lim .2 + 1
> Explain in your own words the meaning of each of the following. (a) lim f(x) = 5 (b) lim f(x) = 3 X00 X -00
> a. Prove that And if these limits exist. b. Use part (a) and Exercise 65 to find lim f(x) = lim f(1/t) lim f(x) lim f(1/t) 1 lim x sin x→0+
> Formulate a precise definition of Then use your definition to prove that lim f(x) = X -00 lim (1 + x³) X -0 8.
> Use Definition 9 to prove that lim e* = 0.
> Determine the infinite limit. x + 1 lim x-5 x - 5
> Prove, using Definition 9, that lim x' = 0.
> Use Definition 8 to prove that 1 0. lim X-0 X
> a. How large do we have to take x so that 1/x2 b. Taking r = 2 in Theorem 5, we have the statement Prove this directly using Definition 7. 1 lim .2
> For the limit illustrate Definition 9 by finding a value of N that corresponds to M = 100. lim vx In x = 00 X00
> For the limit illustrate Definition 8 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 — 3x lim = 3 -2 x² + 1 x- X -00
> For the limit illustrate Definition 7 by finding values of N that correspond to ε = 0.1 and ε = 0.05. 1 – 3x lim Vx2 + 1 -3
> Use a graph to find a number N such that 3x? + 1 if x> N then 1.5 < 0.05 | 2x2 + x + 1
> a. By graphing y = e-x/10 and y = 0.1 on a common screen, discover how large you need to make x so that e-x/10 < 0.1. b. Can you solve part (a) without using a graphing device?
> In Chapter 9 we will be able to show, under certain assumptions, that the velocity v(t) of a falling raindrop at time t is v(t) = v*(1 - e-gt/v*) Where g is the acceleration due to gravity and v* is the terminal velocity of the raindrop. a. Find limt →
> Determine the infinite limit. x + 1 lim X-5+ x - 5
> 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 L/min. Show that the concentration of salt after t minutes (in grams per liter) is C(t) = 30t/200 + t b. What happens to
> Find limx → ∞ f(x) if, for all x > 1, 10e* – 21 ) < I – x^ /x – 1 <f(x) 2e*
> By the end behavior of a function we mean the behavior of its values as x → ∞ and as x → -∞. a. Describe and compare the end behavior of the functions P(x) = 3x5 - 5x3 + 2x Q(x) = 3x5 by graphing both functions in the viewing rectangles [-2,
> a. Use the Squeeze Theorem to evaluate b. Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote? sin x lim
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x+(x2 – 1)-(x + 2)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = (3 - x)(1 + x)2(1 - x)4
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x3(x + 2)2(x - 1)
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = x4 - x6
> Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12. y = 2x3 - x4
> A function f is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one x-intercept, x = 1. It is known that f has a removable discontinuity at x = -1 and limx →-1 f(x) = 2. Evaluate a. f(0) (b) lim f(x)
> a. Estimate the value of by graphing the function f(x) = (sin πx)/(sin πx). State your answer correct to two decimal places. b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0. sin x lim X0 si
> The point Ps2, 21d lies on the curve y = 1/(1 – x). a. If Q is the point (x, 1/(1 - x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x: i. 1.5 ii. 1.9 iii. 1.99 iv. 1.999 v
> Find a formula for a function that has vertical asymptotes x = 1 and x = 3 and horizontal asymptote y = 1.
> Find a formula for a function f that satisfies the following conditions: lim f(x) = 0, lim f(x) = -, f(2) = 0, %3D lim f(x) = ∞, lim f(x) = -00
> Make a rough sketch of the curve y = xn (n an integer) for the following five cases: i. n = 0 ii. n > 0, n odd iii. n > 0, n even iv. n v. n Then use these sketches to find the following limits. (а) lim x^ X0+ (b) lim x" X0- (c) lim x" (d)
> Let P and Q be polynomials. Find if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q. P(x) lim Q(x)
> a. Graph the function f(x) = 2x2 + 1 /3x - 5 How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits b. By calculating values of f(x), give numerical estimates of the limits in part (a). c. Cal
> Estimate the horizontal asymptote of the function f(x) = 3x3 + 500x2/x3 + 500x2 + 100x + 2000 by graphing f for -10 ≤ x ≤ 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
> 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/ ex - 5
> 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 = x3 – x/x2 - 6x + 5
> 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 + x4/x2 - x4
> 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 = 2x2 + x – 1/x2 + x - 2
> a. By graphing the function f(x) = (cos 2x - cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx→0 f(x). b. Check your answer in part (a) by evaluating f(x) for values of x that approach 0.
> 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 = 2x2 + 1/3x2 + 2x - 1
> 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 = 5 + 4x/x + 3
> a. Use a graph of f(x) = 3x2 + 8x + 6 - 3x2 + 3x + 1 to estimate the value of limx→∞ f(x) to one decimal place. b. Use a table of values of f(x) to estimate the limit to four decimal places. c. Find the exact value of the limit.
> a. Estimate the value of by graphing the function f(x) = x2 + x + 1 + x. b. Use a table of values of f(x) to guess the value of the limit. c. Prove that your guess is correct. lim (Vx² + x + 1 + x) X -0
> For f(x) = 2/x – 1/ln x find each of the following limits. e. Use the information from parts (a)–(d) to make a rough sketch of the graph of f. (a) lim f(x) (b) lim f(x) X→0+ (c) lim f(x) (d) lim f(x) x→1- x→1+
> a. For f(x) = x/ln x find each of the following limits. b. Use a table of values to estimate c. Use the information from parts (a) and (b) to make a rough sketch of the graph of f. (i) lim f(x) (ii) lim f(x) (iii) lim f(x) X→0+ I→I+ lim f(x).
> Find the limit or show that it does not exist. lim [In(2 + x) – In(1 + x)]
> Find the limit or show that it does not exist. lim [In(1 + x²) – In(1 + x)]
> Find the limit or show that it does not exist. lim tan (In x)
> Find the limit or show that it does not exist. cos x) -2x lim (e
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim x' Inx -2
> Find the limit or show that it does not exist. 1 — е* lim x0 1 + 2e*
> Find the limit or show that it does not exist. e t - e lim x0 e 3x + e -3x — е
> Find the limit or show that it does not exist. lim arctan(e*)
> Find the limit or show that it does not exist. 1 + x* lim x* + 1 x-0
> Find the limit or show that it does not exist. lim (x² + 2x²) X-0
> Find the limit or show that it does not exist. lim (e* + 2 cos 3x)
> Find the limit or show that it does not exist. x* – 3x? + x lim x + 2
> Find the limit or show that it does not exist. lim Vx2 + 1
> Find the limit or show that it does not exist. lim (Vx2 + ax Vx² + bx
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim x*
> Find the limit or show that it does not exist. lim (V4x2 + 3x + 2x)
> Find the limit or show that it does not exist. lim (/9x? + x – 3x)
> Find the limit or show that it does not exist. x + 3x² lim x0 4x - 1 .2
> Find the limit or show that it does not exist. Vx + 3x? lim 4х — 1
> Find the limit or show that it does not exist. V1 + 4x6 lim .3 X -0 2 - x
> Find the limit or show that it does not exist. V1 + 4x6 lim .3 2 -
> Find the limit or show that it does not exist. x2 lim /x* + 1
> Find the limit or show that it does not exist. (2x² + 1)² lim (x – 1)(x² + x)
> Find the limit or show that it does not exist. lim a 2t3/2 + 3t – 5
> Find the limit or show that it does not exist. VE + t? lim 2t – t?
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 5' – 1 lim 0 t
> Find the limit or show that it does not exist. 4x3 + 6х2 — 2 lim 2x3 — 4х + 5 X-00
> Find the limit or show that it does not exist. x - 2 lim x-0 x2 + 1
> Find the limit or show that it does not exist. 1 - x? .2 lim .3 x→* x' - x + 1
> Find the limit or show that it does not exist. Зх — 2 lim х 2х + 3x >00
> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 9x³ + 8x — 4 lim 3 — 5х + x3
> Evaluate the limit and justify each step by indicating the appropriate properties of limits. 2x? – 7 lim 5x? + x – 3
> a. Use a graph of f(x) = (1 – 2/x)2 to estimate the value of limx→∞ f(x) correct to two decimal places. b. Use a table of values of f(x) to estimate the limit to four decimal places.
> Guess the value of the limit by evaluating the function f(x) = x2/2x for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess. .2 lim X→* 2*
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) lim f(x) = 2, f(0) = 0, f is even -00, x→3
> Sketch the graph of an example of a function f that satisfies all of the given conditions. f(0) = 3, lim f(x) = 4, x>0- lim f(x) = 2, x→0+ lim f(x) lim f(x) x→4- lim f(x) -00, -00, 00, X -00 X→4+ lim f(x) = 3
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. sin 30 S lim 00 tan 20
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = 3, lim f(x) = ∞, lim f(x) -00, ƒ is odd x→2- x→2+
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) - 0o, lim f(x) = ∞, lim f(x) = 0, x→2 X00 X -00 lim f(x) x0+ / (X) :
> Sketch the graph of an example of a function f that satisfies all of the given conditions. lim f(x) = ∞, x→2 lim f(x) = ∞, x→-2+ lim f(x) = x→-2- im J(x) = 0, lim f(x) = 0, f(0) = 0 X -00
> a. From the graph of f, state the numbers at which f is discontinuous and explain why. b. For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither. y A -2 4 6. 2.
> If f is continuous on (-∞, ∞), what can you say about its graph?
> Write an equation that expresses the fact that a function f is continuous at the number 4.
> A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the In
> a. Show that the absolute value function F(x) = |x| is continuous everywhere. b. Prove that if f is a continuous function on an interval, then so is | f |. c. Is the converse of the statement in part (b) also true? In other words, if | f | is continuou
> Show that the function is continuous on (-∞, ∞) x* f(x) = sin(1/x) if x 0 if x = 0
> If a and b are positive numbers, prove that the equation a/ x3 + 2x2 – 1 + b/ x3 + x - 2 = 0 has at least one solution in the interval (-1, 1).
> Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 1 + p° lim p1 1+ p5
> Is there a number that is exactly 1 more than its cube?
> For what values of x is g continuous? 0 if x is rational g(x) x if x is irrational
> For what values of x is f continuous? |0 if xis rational f(x) 1 if x is irrational
> a. Prove Theorem 4, part 3. b. Prove Theorem 4, part 5.
> Prove that cosine is a continuous function.
> To prove that sine is continuous, we need to show that limx→a sinx = sina for every real number a. By Exercise 63 an equivalent statement is that Use (6) to show that this is true. lim sin(a + h) = sin a
> Prove that f is continuous at a if and only if lim f(a + h) = f(a) h→0
> Prove, without graphing, that the graph of the function has at least two xintercepts in the specified interval. y = x2 - 3 + 1/x, (0, 2)
