Limits and continuity #1 - Questions and Answers

Question 1 Which of the following is NOT a result of the Central Limit Theorem? Choose the correct answer below. a) As the sample size increases, the distribution of the sample means x-bar will approach a normal distribution. b) The distribution of the sample data will approach a normal distribution as the sample size increases. c) The variance of all sample means is the population variance divided by the square root of the sample size. d) The mean of all sample means is the population mean μ Answer Step 1 The options below are not correct As the sample size grows, the distribution of the sample mean becomes more normal. The standard deviation of all sample means is the ratio of the population standard deviation to the square root of sample size, moreover the expected value of all sample means is equal to the population mean. When the sample size sufficiently large then the distribution of sample mean leads to normal and the expected value of all sample means equal to population mean. The standard deviation of all sample means is equal to the population standard deviation divided by the square root of sample size. Step 2 The correct option is identified below: When sample size increases the distribution of sample data will not follow normal distribution but the average of sample mean leads normal. The distribution of the sample data will approach a normal distribution as the sample size increases is not a conclusion of central limit theorem. According to central limit theorem the distribution of sample data does not follow normal distribution as the sample size increases or decreases but the average of sample mean follows normal distribution. Question 2 Each limit represents the derivative of some function f at some number a. Indicate an f and a in each instance. lim t→1 (t⁴ - t - 2) / (t - 1) Answer 1 Step 1 Consider the limit lim t→1 (t⁴ - t - 2) / (t - 1) We want to represent it as a derivative of some function f at some number a, In light of definition 5 we want to write our limit in the form lim t→a (f(t) - f(a)) / (t - a) Since t - 1 appears in the denominator, and t→ 1, we are motivated to take a = 1 Looking at the numerator of the limit in question we want f(t) = t⁴ + t. Let us see if this is the right choice. By definition 5 we have f'(1) = lim t→1 (f(t) - f(1)) / (t - 1) = lim t→1 (t⁴ + t - (1⁴+1)) / (t - 1) = lim t→1 (t⁴ + t - 2) / (t - 1) Which is what we wanted to establish. Ans: f(t) = t⁴ + t, a = 1 Answer 2 Step 1 Definition f'(a) = lim x→a (f(x) - f(a)) / (x - a) Step 2 lim t→1 (t⁴ + t - 2) / (t - 1) lim t→1 (t⁴ + t - (1⁴+1)) / (t - 1) lim t→1 (f(t) - f(1)) / (t - 1) = f'(1) Where f(t) = t⁴ + t Note that a = 1 Answer 3 To find the function and number associated with the given limit lim t→1 (t⁴ - t - 2) / (t - 1), we can rewrite it using the derivative notation. Let's start by simplifying the expression: lim t→1 (t⁴ - t - 2) / (t - 1) Factoring the numerator as (t - 2)(t + 1)(t² + 1), we can cancel out the common factor of (t - 1) in both the numerator and denominator: lim t→1 ((t - 2)(t + 1)(t² + 1)) / (t - 1) Now, we can see that the limit resembles the derivative of a function. To find the function f and number a associated with this limit, we rewrite it as: lim t→1 (t⁴ - t - 2) / (t - 1) = lim t→1 (t + 1)(t² + 1) By observing the expression inside the limit, we can identify the function f(t) = t⁴ + 1 and the number a = 1. Therefore, the given limit represents the derivative of the function f(t) = t⁴ + 1 at t = 1. Answer 4 Explanation: lim t→1 (t⁴ - t - 2) / (t - 1) To simplify this expression, we can factor the numerator using the difference of cubes formula: t⁴ - t - 2 = (t² + 1)(t² - 2) Now we can rewrite the limit as: lim t→1 ((t² + 1)(t² - 2)) / (t - 1) Since the denominator is t - 1, we can cancel it out with the factor (t² - 2) in the numerator: lim t→1 (t² + 1)(t + 1) Next, we can evaluate the limit as t approaches 1. Plugging in t = 1 into the expression, we get: (1² + 1)(1 + 1) = 2(2) = 4 Therefore, the function f(t) that represents the given limit is f(t) = t⁴ + t, and the number a is a = 1. Answer 5 Given limit: lim t→1 (t⁴ - t - 2) / (t - 1) To simplify the expression, we can factorize the numerator: ((t - 2)(t + 1)(t² + 2t + 1)) / (t - 1) Now, cancel out the common factor of (t - 1) from the numerator and the denominator: (t - 2)(t + 1)(t + 1) Simplifying further, we get: (t - 2)(t + 1)(t + 1) Now, let's find a function f(t) whose derivative matches this expression. We can take: f(t) = (t - 2)³/3 To find the value of a, we substitute t = 1 into the function: f(1) = (1 - 2)³/3 = -1/3 Therefore, the function f(t) and the value of a for the given limit are: f(t) = (t - 2)³/3 and a = 1 So, the solution can be written as: lim t→1 (t⁴ - t - 2) / (t - 1) = lim t→1 (t - 2)(t + 1)(t + 1) = f'(1), where f(t) = (t - 2)³/3 and f'(t) represents the derivative of f(t). Question 3 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x) = 1/(x + 2) if x ≠ -2 , 1 if x = -2 a = -2 Answer A function f is continuous from the right at x=a if lim x→a+ f(x) = f(a) And f is continuous from the left at x=a if lim x→a- f(x) = f(a) f is said to be continuous when it is continuous from both left and Right. f(x) = 1/(x + 2) if x ≠ -2, 1 if x = -2 Graph for f(x) is shown below: (graph) From the graph we can see that: lim x→-2- f(x) = -∞ And lim x→-2+ f(x) = +∞ But f(-2) = 1 Therefore f is discontinuous at x = -2 from both left and Right. Result: f is discontinuous at x=-2 from both left and Right Question 4 If 4 ≤ xg(x) ≤ 2x⁴ – 2x² + 4 for all x, evaluate lim x→1 g(x) Answer Given, 4x ≤ xg(x) ≤ 2x⁴ - 2x² + 4 for all x. Now, lim x→1 4x = 4(1) = 4 and lim x→1 (2x⁴ - 2x² + 4) = 2(1)⁴ - 2(1)² + 4 = 2 - 2 + 4 = 4 Question 5 How do you find the limit as x approaches infinity of y = arctan(e^x)? Answer As lim x→+∞ e^x = +∞ and lim x→+∞ arctan(x) = π/2 Thus, if x → ∞, e^x → ∞ so that, letting t = e^x, you have: lim x→+∞ arctan(e^x) = lim t→+∞ arctan(t) = π/2