Limits & Continuity Questions and Answers

Find the following limits for the rational function f(x)=x+2/(4x² -16) and justify your answers. If the limit does not exist, state so and show or explain why it does not exist. (a) lim f(x) = x➡ -2 (b)lim f(x) = x ➡2 (c)lim f(x) = x➡∞

Which of the following is a solution to the equation sin²x - cos²x = 1? (A) x = 0 (B) x = π/6 (C) x = π/4 (D) x = π/2 (E) x = π

Which of the following statements is NOT true about the curve defined by the parametric equations x(t) = 12 cos πt and y(t) = 12 sinπt? (A) The curve is centered at the origin (B) The path of the curve can be represented by the polar equation r = 12 (C) A particle following the path of the curve will be located at the point(0,12)at t=0 (D) A particle following the path of the curve will travel counterclockwise as t increases (E) A particle following the path of the curve will be at the same point along the curve at t = 0 and t = 2

If lim f(x) = 5 and lim g(x) = 3, then lim [g(x) = f(x)] = (A)- 2 (B) 0 (C) 2 (D) 3 (E) 5

Sebastian invested $7,100 in an account paying an interest rate of 81% compounded monthly. Eva invested $7,100 in an account paying an interest rate of 81% compounded continuously. After 5 years, how much more money would Eva have in her account than Sebastian, to the nearest dollar?

A petroleum company is making a cylindrical oil drum. The drum must hold 1.9 m³ of liquid. The drum must be taller than it is wide, but no more than 1.7 m tall. What are the dimensions of the drum that uses the least amount of material?

Find the first five terms of the sequence of partial sums. (Round your answers to four decimal places.) ∑ (-5)ⁿ+¹ s₁= s₂= s₃= s₄= s₅=

Let h(x) = 2x - x³. Use the limit definition of the derivative to differentiate h. h'(x) = ____________ Determine the slope of h at x = -3. h'(-3) =

The limit below represents f'(c) for some function f and real number c. Iim [-1+(-2+Δx)^2+9]/Δx Δx→0 a)Determine the function f. f(x) = b)Determine the value of c. C=

Consider the following repeating decimal. (a) Write the repeating decimal as a geometric series. n=0 (b) Write the sum of the series as the ratio of two integers.

In which quadrant does θ lie if the following statements are true: sin θ > θ and cos θ > θ A)Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV

Evaluate the following limits. Justify each step. If the limit does not exist, explain why. a) lim 2ln|x + 2| − ln(x^4 + 4x³ + 4x²) b) lim sin(5x) csc (7x) c) lim(16x^3-2x^2+1)^(1/2)/(1+x^6)^(1/4)

A rectangular paperboard measuring 30 in long and 20 in wide has a semicircle cut out of it, as shown below. What is the perimeter of the paperboard that remains after the semicircle is removed?

Find the nth Taylor polynomial for the function, centered at c. f(x) =1/x² n = 4, c = 2 P₄(x) =

Martha Stewart is designing a box for her cat to sleep in. It has 4 sides and a bottom but no top. The plush material for the square bottom of the box costs $18/ft² and the material for the sides costs $12/ft². She needs a box with a volume of 18 ft^ 2. Find the dimensions of the box that minimize the cost.

Find the distance between the pair of points (3.3,7.2) and (-0.7,5.2). If numbers places. The distance between the given points is ________ units.

Use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). (Round your answers to three decimal places.) y = √2x upper sum _______ lower sum________

Use the Integral Test to determine the convergence or divergence of the p-series. ∑ 1/n⁴ ∫₁1/n⁴ (a) converges (b) diverges

Determine if the following series are convergent or divergent. Justify your steps and state which test you are using. When necessary, make sure you check the hypotheses of the test that are satisfied before you apply it. (a)Σ (-1)^n*(1/n^n) (b)Σ 6^n/(5^n+8) (c)Σn^3/(2n^4+3n+2) (d)Σn!/(n+2)! (e)Σcos(πn+5)/3^n

Decide whether the integral is improper. ∫ sin(x)/26+x² dx (a)proper (b)improper Explain your reasoning. (Select all that apply.) (a)The limits of integration are both finite. (b)The integrand is continuous on (-∞, ∞). (c)The integrand is not continuous on (-∞, ∞). (d)At least one of the limits of integration is not finite.

Write as a single logarithm. Simplify. 2 log₃ u- log₃ v =

Evaluate the following integrals. (a) ∫(4x + 1)/(x-2)(x - 3)² dx (b) (√x)ln(√x) dx (c)(x⁵ + x + 1)/(x² + 2)dx

During the design phase of one of its model spacecraft, Spacez launches the Atlas 31415 rocket vertically. A camera is positioned 5000 ft from the launch pad. When the rocket is 12,000 feet above the launch pad, its velocity is 800 ft/sec. Find the necessary rate of change of the camera's angle as a function of time so that it stays focused on the rocket.

For the series problems, show ALL conditions required for the specified test for convergence or divergence. The Limit Comparison Test Σ₁ ∞ 2/4n+1

Jackson invested $590 in an account paying an interest rate of 61% compounded monthly. Ellie invested $590 in an account paying an interest rate of 61% compounded continuously. After 19 years, how much more money would Ellie have in her account than Jackson, to the nearest dollar?

Gabriel invested $77,000 in an account paying an interest rate of 3 1/2% compounded monthly. Isabella invested $77,000 in an account paying an interest rate of 41/8% compounded quarterly. After 6 years, how much more money would Isabella have in her account than Gabriel, to the nearest dollar?

Evaluate exactly: (a) log₃27+ log₇ (b) In √e= (c) 2e^ln2=

Its radius is 38 m. Find the circumference and area of the courtyard. Circumference:_________ Area:________________

Simplify as instructed. (1) sec(t) — cos(t)= _________ Using sec(t) = 1 / cos(t) (2) = _________ adding using common denominator (3) = ____________ Using sin²(t) + cos² (t) = 1

Let h(x) = -9 - 2x. Use the limit definition of the derivative to differentiate h. h'(x) = _ ______________ Determine the slope of h at x = 0. h'(0) =_______________

Let g(x) = x² + 3x +5 Use the alternative limit definition of the derivative to determine the derivative (slope) of g at x = 2. g'(2) = _____________

(1) Multiply. (1 + cos(t))(1 − cos(t))= __________________ (2) Simplify further using identity sin² (t) + cos² (t) - 1 (1 + cos(t)) (1 - cos(t))= ___________

For y = 6 cos(1/2)x (1) it's amplitude is __________ (2) it's period is _________

The following equation is_______ true. 2sin(x)=√₂ (A) Always (B) Never (C) Not enough information (D) Sometimes The following equation is_______ true cos(x) = √3 (A) Never (B) Sometimes (C) Always (D) Not enough information

Let f(x) = {√x+1 if x<3 {5- -x if x ≥ 3. Determine the one-sided limits as x approaches 3, and determine the limit as x approaches 3.

Given x = t² + 2t - 1 and y = t² + 4t + 4, what is the equation of the tangent line at t = 1.

Determine whether each of the following geometric series converges or diverges. If it converges, find its sum. (a)∑^∞n=1 36ⁿ/2²ⁿ+¹ (b) ∑^∞n=3 4ⁿ3-²ⁿ-²

Of all points (x ,y ,z) that satisfy x + 4y + 3z=21, find the one that minimizes (x − 1)² + (y − 1)² + (z − 1)².The given function has a minimum at (0,0,0) which satisfies x + 4y + 3z = 21.

Solve for x: a) 3ˣ² = 9ˣ b) 3¹-²ˣ = 4ˣ c) log₃ (4x - 7) = 2 d) log₄(x + 3) + log₄(2-x) = 1

Of all the numbers whose sum is 20, find the two that have the maximum product. The two numbers whose sum is 20 and that have the maximum product are.... Simplify your answer. Use a comma to separate answers as needed.

Find the extremum of f(x,y) subject to the given constraint, and state whether it is a maximum or a minimum. f(x,y) = xy; 14x + y = 14 There is a ________ value of ________ located at (x ,y)=

Find and simplify the expression if f(x)=x²-9 f(4+h)-f(4)=_______

Question 8 Solve: log_3(9x + 17) – log_3( – 5x - 15)=2 x=_____ (Enter DNE if no solution exists)

Prove the following propositions for all positive integers n. (a) 1+5+9+ 13 +...(4n - 3)=n(4n-2)/2 (b)Σk(k=1,n)=n(n+1)/2

(05.01, 5.05, 5.07 MC) Let θ = - (8π/3) Part A: What is a coterminal angle of θ such that 0 ≤ θ ≤ 2π? Part B: What are the exact values of all six trigonometric functions evaluated at θ?

This problem has two parts. Part 1: Given that y = 4 sin(α), with α € (0,π/2), express each of the following as a function of y. cos(α) cos(α) cot(α) = tan(α)² =

Use the graph of f to find the indicated limit and function value. a. lim f(x) b. f(-2) a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A.lim f(x) = x->-2 (Type an integer or a decimal.) B. The limit does not exist.

Solve: (4x - 21)/ (x - 6) ≤ 3 x ∈ ___________ (Enter your answer in INTERVAL notation, using U to indicate a union of intervals; or enter DNE if no solution exists)

Use the equation for the function f(x) = x³ - 10 to find and graph the further Find f¯¹(x). f-¹(x) =_____

Orange juice, a raisin bagel, and a cup of coffee from Billy's Breakfast Bar cost a total of $3.00. Billy posts a notice announcing that, effective the following week, the price of orange juice will increase 50% and the price of bagels will increase 20%. After the increase, the same purchase will cost a total of $3.75, and the orange juice will cost twice as much as coffee. Find the price of each item before the increase. (A) What was the cost of a glass of orange juice before the increase? (B) What was the cost of a raisin bagel before the increase? (C) What was the cost of a cup of coffee before the increase?