Concept explainers
Limit proof Use the formal definition of a limit to prove that
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Calculus: Early Transcendentals, 2nd Edition
Additional Math Textbook Solutions
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus and Its Applications (11th Edition)
University Calculus: Early Transcendentals (3rd Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
- Demostrate thatarrow_forwardFigure out that this statement is true or false? if is false explain why? by using example, and if it is true explain why? When lim x → a f ( x ) exists, the limit is always equal to f ( a ) - Is this statement true or false?arrow_forwardAssume that lim f(x, y) = 3 (x,y)→(2,5) lim g(x, y) = 4 (x,y)-(2,5) Evaluate the following limit. lim (x,y)-(2,5) ef(x,y)²-6g(x,y) (Use symbolic notation and fractions where needed.) limef(x,y)²-6g(x,y) – = (x,y)-(2,5)arrow_forward
- Calculusarrow_forwardI attached the imagearrow_forward" (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…arrow_forward
- x²+1 Evaluate the limit of limx-→∞( -- ax – b) = 0 x+1arrow_forwardn Exercises 1-12, evaluate the indicated limit or explain why it does not exist. sin(xy) 2x2 - ху 9. lim 10. lim (r,y)-(0,0) x2 + y2 (x.y)(1,2) 4x2-y2 1. lim xy +x2 2. Vx2 + y2 lim (x.y) (2,-1) 11. lim (r,y)(0,0) x2 + v4 13. Kow can the functionT (x.y) (0,0) lim (r,y)(0,0) 2r4+ y4 x² + y? 3. lim 4. lim x2 +y2-x'y3 (x.y) (0,0) y (x.y) (0,0) x2 + y? f(x.y)%D (x. y) # (0.0). x2 + y2 cos(xy) x(y-1)2 be defined at the origin so that it becomes continuous at all points of the xy-plane? 5. lim 6. X.y)(I.) 1-XIcos y lim (x.y) (0.1) x2 +(y-1)2 14. How can the function 7. sin(x - y) x-y lim (r.y) (0,0) X2+ y2 8. lim (x.y)-(0,0) Cos(x + y) f(x, y) = (x#y). x-yarrow_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_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning