
bonse. Question 5 Suppose that f(x) nx2 1 + nx?' for all XER. What is lim...
Question 3 Suppose that f(x) nx2 1 + nx?' for all x ER. What is fo) for all nEN.
(1) If f: R₃ R a continuous function such that f(x)² > 0 for all xER. Show that either f(x) >0 for all a ER or f(x) <0 all X E R.
XL Xa 12. (a) Suppose that f(x) = g(x) for all x. Prove that lim f(x) < lim g(x), provided that these limits exist. (b) How can the hypotheses be weakened? (c) If f(x) < g(x) for all x, does it necessarily follow that lim f (x) < lim g(x)? Ya X-
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
3. Given that lim f(x) = 4 and lim f(x) = -4, what is true about lim f(x)? x-1 4. Given that lim f(x) = 1, lim f(x) = 1, and f(-3) = 1, what is true about lim f(x)? 5. Given that lim f(x) = 0, lim f(x) = 0, and f(2) = 3, what is true about limf(x)? X-3 X-3 X-2
1. Suppose that lim x) = A, lim f(x) = B, 0) = C, where A, B, C are distinct real numbers. In each of the following, fill in the corresponding box by: • Expressing the limit in terms of A, B, C if it is possible to do so using the given information; • Writing DNE if it is possible to conclude that the limit does not exist using the given information; • Putting a X. otherwise. No explanation...
1 Given lim f(x) = 5, lim g(x) = -2, and lim h(x) = 0, evaluate the limit: lim x-c f(x) - 8(x) 1-C X- X-C 1 5 b) c) d) e) does not exist
5. Suppose and g are two functions continuous at x = 1. If lim and lim g(x) = 10, find f(1) and g(1). Justify your solution. mous at z = 1. If bug (in (52) s(e) * - ) = 21 6. Show that each of the following equations has a solution: a) sin(a) + x3 = 0. b) In(a)e* + 1*22 = 0).
1. (6) Sketch the graph f a function f that satisfies all of the given conditions. lim )3, im ()5, lim , lim+ =-oo.lim2--5, f is continuous from the left at x--1. 2
1. (6) Sketch the graph f a function f that satisfies all of the given conditions. lim )3, im ()5, lim , lim+ =-oo.lim2--5, f is continuous from the left at x--1. 2
1. Evaluate lim f(x) if 1 – x4 = f(x) < cosx for all x in X+0 2. Find f'(x) when (a) f(x) = cscZx - cotax (b) f(x) = Vă -