Find the limit. limx→2+ e3/ (2 – x)
> (a). Find the slope of the tangent to the curve y = 3 + 4x2 - 2x3 at the point where x = a. (b) Find equations of the tangent lines at the points (1, 5) and (2, 3). (c). Graph the curve and both tangents on a common screen.
> Find an equation of the tangent line to the curve at the given point. y = 2x + 1/x + 2, (1, 1)
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Determine lim x→1- 1/x3 – 1 and limit x→1+ 1/x3 – 1 (a). by evaluating f (x) = 1/ (x3 – 1) for values of that approach 1 from the left and from the right, (b). by reasoning as in Example 1, and (c). from a graph of f.
> (a). Can the graph of y = f (x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b). How many horizontal asymptotes can the graph of y = f (x) have? Sketch graphs to illustrate the possibilities.
> Find an equation of the tangent line to the curve at the given point. y = x3 - 3x + 1, (2, 3)
> Find an equation of the tangent line to the curve at the given point. y = 4x - 3x2 (2, -4)
> For the function whose graph is given, state the following. (a). lim x→∞ g (x) (b). lim x→-∞ g (x) (c). lim x→3 g (x) (d). lim x→0 g (x) (e). l
> For the function f whose graph is given, state the following. (a). limx→2 f (x) (b). limx→-1- f (x) (c). limx→-1+ f (x) (d). limx→∞ f (x) (e). lim xâ†
> Graph the curve y = ex in the viewing rectangles [-1, 1] by [0, 2], [-0.5, 0.5] by [0.5, 0.5], and [-0.1, 0.1] by [0.9, 1.1. What do you notice about the curve as you zoom in toward the point (0, 1)?
> A curve has equation y = f (x). (a). Write an expression for the slope of the secant line through the points P (3, f (3)) and Q (x, f (x). (b). Write an expression for the slope of the tangent line at P.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> In the theory of relativity, the mass of a particle with velocity is where m0 is the mass of the particle at rest and is the speed of light. What happens as v→c-?
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.
> Find limx→∞ f (x) if, for all x > 1,
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> To prove that sine is continuous we need to show that limx→a sin x = sin a for every real number a. If we let h = x - a, then x = a + h and x→a ⇔ h → 0. So, an equivalent statement is that limh→a sin (a + h) = sin a Use (6) to show that this is true.
> If limx→1 f (x)/x2 = 5 find the following limits. (a). limx→0 f (x) (b). limx→0 f (x)/x
> Estimate the horizontal asymptote of the function f (x) = 3x2 + 500 x2/ x2 + 500 x2 + 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?
> (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.
> 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.
> If f (x) = [[x]] + [[-x]], show that limx→2 f (x) exists but is not equal to f (2).
> 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.
> Guess the value of the limit x→∞ x2/2x 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.
> 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.
> (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 lim (b). By calculating values of f, give numerical estimates of the limits
> Find the limit. limx→∞ x + x3 + x5/1 – x2 + x4
> Find the limit. limx→(π/2)+ etanx
> Find the limit. limx→-∞ (x4 + x5)
> Find the limit. limx→∞ e3x – e-3x/ e3x + e-3x
> Find the limit. limx→∞ (e-2x cos x)
> Find the limit. limx→∞ sin2x/x2
> Find the limit. limx→∞ cos x
> Find the limit. limx→∞ e√x2 + 1
> A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. (a). If P is the point (15, 250) on the graph of V, fin
> Find the limit. limx→∞ e-x2
> Find the limit. limx→∞ (√x2 + ax – √x2 + bx
> Find the limit. limx→∞ (√9x2 + x – 3x)
> Find the limit. Limu→∞ 4u4 + 5/ (u2 – 2) (2u2 – 1)
> Find the limit. limx→-∞ t2 + 2/ t3 + t2 – 1
> Find the limit. limx→∞ x3 + 5x/ 2x3 – x2 + 4
> Find the limit. limx→2- x2 – 2x/ (x2 – 4x + 4
> If f is continuous on (-∞, ∞), what can you say about its graph?
> Find the limit. limx→3+ ln (x2 – 9)
> Find the limit. limx→π cot x
> (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.
> Find the limit. limx→-3- x + 2/ x + 3
> Find the limit. limx→1 2 - x/ (x – 1)2
> Use a graph to estimate all the vertical and horizontal asymptotes of the curve y = x3/x3 - 2x + 1
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Write an equation that expresses the fact that a function is continuous at the number 4.
> A parking lot charges $3 for the first hour (or part of an hour) and $2 for each succeeding hour (or part), up to a daily maximum of $10. (a). Sketch a graph of the cost of parking at this lot as a function of the time parked there. (b). Discuss the disc
> Sketch the graph of a function that is continuous except for the stated discontinuity. Neither left nor right continuous at -2, continuous only from the left at 2
> Sketch the graph of a function that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5
> Sketch the graph of a function that is continuous except for the stated discontinuity. Discontinuities at -1 and 4, but continuous from the left at -1 and from the right at 4
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function.
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).
> Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Sketch the graph of an example of a function f that satisfies all of the given conditions.
> Prove that cosine is a continuous function.
> Sketch the graph of a function that is continuous except for the stated discontinuity. Discontinuous, but continuous from the right, at 2
> If limx→1 f (x) – 8/x-1 = 10, find limx→1 f (x).
> If r is a rational function, use Exercise 43 to show that limv→a r (x) = r (a) for every number a in the domain of r. Exercise 43: If p is a polynomial, show that limv→a P (x) = P (a).
> If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after seconds is given by H = 10t – 1.86t2. (a). Find the velocity of the rock after one second. (b). Find the velocity of the rock when t = a. (c). When will
> If p is a polynomial, show that limv→a P (x) = P (a).
> In the theory of relativity, the Lorentz contraction formula expresses the length L of an object as a function of its velocity v with respect to an observer, where l0 is the length of the object at rest and c is the speed of light. Find limv→c-L and inte
> Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why.
> From the graph of f, state the intervals on which is continuous.
> (a). If the symbol [[]] denotes the greatest integer function defined in Example 9, evaluate (b). If n is an integer, evaluate (c). For what values of does limx→a [x] exist?
> Suppose that a function f is continuous on [0, 1] except at 0.25 and that f (0) = 1 and f (1) = 3. Let N = 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f mi
> Let (a). Evaluate each of the following, if it exists. (b) Sketch the graph of g.
> Find the limit, if it exists. If the limit does not exist, explain why.
> Find the limit, if it exists. If the limit does not exist, explain why.
> The gravitational force exerted by the earth on a unit mass at a distance r from the center of the planet is where M is the mass of the earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?
> The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.
> Find the limit, if it exists. If the limit does not exist, explain why.
> Show that f is continuous on (-∞, ∞).
> Prove that limx→0 x4 cos 2/x = 0.
> Use continuity to evaluate the limit. limx→3 (x3 – 3x + 1)-3
> Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).
> Use continuity to evaluate the limit. limx→π ex2-x
> Use continuity to evaluate the limit. limx→π sin (x + sin x)
> Use the Squeeze Theorem to show that limx→0 (x2 cos 20πx) = 0. Illustrate by graphing the functions f (x) = -x2, g (x) = x2 cos 20πx, and h (x) = x2 on the same screen.
> (a). Use a graph of f (x) = √3 + x - √3/x to estimate the value of limx→0 f (x) to two decimal places. (b). Use a table of values of f (x) to estimate the limit to four decimal places. (c). Use the Limit Laws to find the exact value of the limit.
> Locate the discontinuities of the function and illustrate by graphing. y = 1/1 + e1/x
> Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie. (a). Describe and compare how the runners run the race. (b). At what time is the distance between the runners the greatest? (c). At what time
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. F (x) = sin (cos (sin x))
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. G (t) = ln (t4 – 1)
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. L (t) = e-5t cos 2πt
> Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.
> Evaluate the limit, if it exists. lim x → -4 1/4 + 1/x /4 + x
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function.
> Explain why the function is discontinuous at the given number a. Sketch the graph of the function.
