Calculus
10th Edition
ISBN: 9781285057095
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 1.3, Problem 111E
To determine
To prove: The expression
and all
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Find a formula for a function f that satisfies the following conditions.
lim f(x) = 0
lim f (x) = -0
f(8) = 0
lim f (x) = oo
lim f (x) = -0
f(-1) =
10
9 - x
f (x) =
22 (x – 8)
-
8 - x
f (x) =
교교 (x - 9)
8 - x
f (x) =
12 (x – 8)
9 - x
f (x) =
2? (x – 9)
O none of these
6) Find two functions f(z) and g(z) such that lim f(z) and lim g(z) do not exist
but lim [f(x) + g()] does exist.
2/2
" (Sum Rule): Suppose f: ℝⁿ → ℝᵐ and g: ℝⁿ → ℝᵐ are functions, and let a ∈ ℝⁿ and b, c ∈ ℝᵐ be points. If lim(x→a) f(x) = b and lim(x→a) g(x) = c, then lim(x→a) (f(x) + g(x)) = b + c.
Proof: Assume that lim(x→a) f(x) = b and lim(x→a) g(x) = c. Let ε > 0 be arbitrary. Then there exists δ₁ > 0 such that for x ∈ Dom(f) with d(x,a) < δ₁, we have ||f(x) - b|| < ε/2 (Equation 1.9). Similarly, there exists δ₂ > 0 such that for x ∈ Dom(g) with d(x,a) < δ₂, we have ||g(x) - c|| < ε/2 (Equation 1.10).
Take δ := min(δ₁, δ₂) and let x ∈ Dom(f + g) satisfy d(x,a) < δ. Since x ∈ Dom(f) and d(x,a) < δ₁, Equation 1.9 holds. Furthermore, x ∈ Dom(g) and d(x,a) < δ₂, so Equation 1.10 applies. We can combine these inequalities:
||f(x) + g(x) - (b + c)|| = ||(f(x) - b) + (g(x) - c)|| ≤ ||f(x) - b|| + ||g(x) - c|| < ε/2 + ε/2 = ε.
This shows that for all x ∈ Dom(f + g) with d(x,a) < δ, we have ||f(x) + g(x) - (b + c)|| < ε. Therefore, f(x) + g(x) → b + c as x → a."
I…
Chapter 1 Solutions
Calculus
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In Exercises 105-110. determine...Ch. 1.4 - Prob. 106ECh. 1.4 - Prob. 107ECh. 1.4 - HOW DO YOU SEE IT? Every day you dissolve 28...Ch. 1.4 - Telephone Charges A long distance phone service...Ch. 1.4 - Prob. 110ECh. 1.4 - Dj Vu At 8:00 a.m. on Saturday, a nun begins...Ch. 1.4 - Volume Use the Intermediate Value Theorem to show...Ch. 1.4 - Prob. 113ECh. 1.4 - Prob. 114ECh. 1.4 - Prob. 115ECh. 1.4 - Signum Function The signum function is defined by...Ch. 1.4 - Prob. 117ECh. 1.4 - Creating Models A swimmer crosses a pool of width...Ch. 1.4 - Making a Function Continuous Find all values of c...Ch. 1.4 - Prob. 120ECh. 1.4 - Prob. 121ECh. 1.4 - Prob. 122ECh. 1.4 - Prob. 123ECh. 1.4 - Prob. 124ECh. 1.4 - Prob. 125ECh. 1.4 - Prob. 126ECh. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Prob. 4ECh. 1.5 - Prob. 5ECh. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Determining Infinite Limits from a Graph In...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Numerical and Graphical Analysis In Exercises...Ch. 1.5 - Prob. 11ECh. 1.5 - Prob. 12ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 14ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 16ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 18ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 20ECh. 1.5 - Prob. 21ECh. 1.5 - Prob. 22ECh. 1.5 - Prob. 23ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 26ECh. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Finding Vertical Asymptotes In Exercises 17-32....Ch. 1.5 - Prob. 29ECh. 1.5 - Prob. 30ECh. 1.5 - Vertical Asymptote or Removable Discontinuity In...Ch. 1.5 - Prob. 32ECh. 1.5 - Prob. 33ECh. 1.5 - Finding a One-Sided Limit In Exercises 3348, find...Ch. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Prob. 36ECh. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Prob. 38ECh. 1.5 - Prob. 39ECh. 1.5 - Prob. 40ECh. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Prob. 42ECh. 1.5 - Prob. 43ECh. 1.5 - Finding a One-Sided Limit In Exercises 37-50, find...Ch. 1.5 - Prob. 45ECh. 1.5 - Prob. 46ECh. 1.5 - Prob. 47ECh. 1.5 - Prob. 48ECh. 1.5 - Prob. 49ECh. 1.5 - Prob. 50ECh. 1.5 - Prob. 51ECh. 1.5 - Prob. 52ECh. 1.5 - Prob. 53ECh. 1.5 - Prob. 54ECh. 1.5 - Prob. 55ECh. 1.5 - Prob. 56ECh. 1.5 - Prob. 57ECh. 1.5 - Relativity According to the theory of relativity,...Ch. 1.5 - Prob. 59ECh. 1.5 - Prob. 60ECh. 1.5 - Rate of Change A 25-foot ladder is leaning against...Ch. 1.5 - Average Speed On a trip of d miles to another...Ch. 1.5 - Numerical and Graphical Analysis Consider the...Ch. 1.5 - Numerical and Graphical Reasoning A crossed belt...Ch. 1.5 - True or False? 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- |x-1| If f (x) = , then lim f(x), lim f(x) and lim f (x) are... respectively x→1+ х—1 x→1- x→1 |x-1| Jika f (x) = , maka lim f(x), lim f(x) dan lim f (x) secara berturut-turut adalah... х-1 x→1+ x→1 1. O 1,-1, tidak ada (does not exist) 2. О -1, -1, -1 3. О 1, 1, 1 4. O -1, 1, tidak ada (does not exist)arrow_forwardShow that if lim f(x) = lim h(x) = 3, and if f(x) < g(x) < h(x) for all x E (-1, 1), then lim g(x) = 3.arrow_forwardLet f(x), g(x) and h(x) be functions, such that complete the blank and check true (V) or false (F).arrow_forward
- find the limitarrow_forward4. Is it possible that there exists some function g(x) and some constant L such that lim g(x) = lim g(x) = L, but lim g(x) does not exist? If so, draw a sketch of such a x-2+ x-2 function. If no, explain why not.arrow_forwardIf lim f(x) = 12 and lim g(x) = 28 then lim[3f(r)g(T) – 8] = O 260 O 1000 O 1008 O24 O 1016arrow_forward
- Let g(x) and h(x) be functions with lim g (x) = G and lim h (x) = H where G and H are real numbers. Which of the following statements is NOT TRUE ? 8 (x) lim h(x) G A H В lim g (x) · h (x) = G ·H lim [8(x)]" = G" lim [s(x) + h(x)]= G + H %3Darrow_forwardLeft- and Right-Hand Limits Not Equal Investigate the one-sided limits of f(x) = as x → 0. Does lim f(x) exist? |x|arrow_forwardLet f and g be real functions. Show directly from the definitionsthat if f is everywhere continuous and limx→1+ g(x) = L, then limx→1+ f(g(x)) = f(L).arrow_forward
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