1. Let's take a look at limits of functions at infinity. Given a function f: R→ R and some LER, we say f(x) converges to L as x → ∞ if for all > 0, there exists some MER such that for all x > M, |f(x) - L| < E In this case, we write f(x) → Las x → ∞, or lim f(x) = L 818 If f does not converge to any LER as x→∞o, we say f diverges as x→∞. (a) Write down a corresponding definition for f(x) to converge to Las x → -∞. Then, use the above definition and the definition you wrote to prove that 1 1 lim = lim xx1+x² 1+x1+x² (b) Suppose f: R→ R satisfies lim f(x)= lim_ f(x) = L 848 for some LE R. Define a function g: R → R by [ƒ(1/y) y‡0 y=0 g(y): = 0 '= Show that g is continuous at 0. (Hint: For the E-d definition of continuity, show that you can break |y| < d into three cases: y = 0, 1/y> 1/8, or -1/y> 1/6. This might help you find the right value of 8.) (c) Continuing from (b), show that if f is continuous at 0, then lim g(y) y4x lim g(y) = f(0) y-1x
1. Let's take a look at limits of functions at infinity. Given a function f: R→ R and some LER, we say f(x) converges to L as x → ∞ if for all > 0, there exists some MER such that for all x > M, |f(x) - L| < E In this case, we write f(x) → Las x → ∞, or lim f(x) = L 818 If f does not converge to any LER as x→∞o, we say f diverges as x→∞. (a) Write down a corresponding definition for f(x) to converge to Las x → -∞. Then, use the above definition and the definition you wrote to prove that 1 1 lim = lim xx1+x² 1+x1+x² (b) Suppose f: R→ R satisfies lim f(x)= lim_ f(x) = L 848 for some LE R. Define a function g: R → R by [ƒ(1/y) y‡0 y=0 g(y): = 0 '= Show that g is continuous at 0. (Hint: For the E-d definition of continuity, show that you can break |y| < d into three cases: y = 0, 1/y> 1/8, or -1/y> 1/6. This might help you find the right value of 8.) (c) Continuing from (b), show that if f is continuous at 0, then lim g(y) y4x lim g(y) = f(0) y-1x
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 8 steps with 8 images
Recommended textbooks for you