Chapter 1.3, Problem 20E

(a) What is wrong with the following equation?   x2 + x − 12 x − 3  = x + 4 (x − 3)(x + 4) ≠ x2 + x − 12 The left-hand side is not defined for x = 0, but the right-hand side is.     The left-hand side is not defined for x = 3, but the right-hand side is.None of these — the equation is correct. (b) In view of part (a), explain why the equation  lim x → 3  x2 + x − 12 x − 3  = lim x → 3 (x + 4)  is correct. Since  x2 + x − 12 x − 3  and x + 4 are both continuous, the equation follows. Since the equation holds for all x ≠ 3, it follows that both sides of the equation approach the same limit as x → 3.     This equation follows from the fact that the equation in part (a) is correct.None of these — the equation is not correct.

Evalute the limit of: lim x->0 (x csc 6x)/(cos 14x) The answer must be in simplified form or a fraction

We know lim 2x -1 = 5 x→3     What value of x guarantees that f(x) = 2x -1 is within 0.2 units of 5?   Show your work.

For the function f(x) = x 4 ln x (a) Show a careful calculation explaining why limx→0+ f(x) = 0 ( asserting that (0)(−∞) = 0 is not careful enough) (b) Calculate a formula for f 0 (x), the derivative of f. (c) Find the critical number(s) for the function f. (d) Determine the absolute maximum and absolute minimum values of f on the interval 1 2 ≤ x ≤ 1, and the exact x values at which these maximum & minimum values occur. Please solve part b c D with a detailed solution.

F(x)={tan(x+3) x<-3           {(x3)3+1 -3<=x   Find lim(x-->3-) f(x) Photo better depicts problem, thank you!

A. Does f(1) exist  B. Does lim x->1 f(x) exist  C. Does lim x->1 f(x) equal f(1) D. Is the function continuous at x=1

A process creates a radioactive substance at the rate of 1 g/hr, and the substance decays at an hourly rate equal to 1/10 of the mass present (expressed in grams). Assuming that there are initially 20 g, find the mass S(t) of the substance present at time t, and find lim S(t) as t approaches infinity

Let G(t) = (1 - cos t)/t2. a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate lim t-->0 G(t). b. Support your conclusion in part (a) by graphing G near t0 = 0

5. Let: f(x) = 1 /x  (a) Use the limit definition of derivative to find f'(x). (b) Verify your answer to (a) using the Power Rule. (c) Use the Power Rule to find f"(x), f'''(x) and f^(4) (x). Find a pattern in the derivatives and make a conjecture for a formula for f^(n) (x), the nth derivative of f(x).