# EXERCISES 2.3 In Exercises 1-4, determine if lim f(x), im f(x), and lim f(x) exist

x-a EXERCISES 2.3 In Exercises 1-4, determine if lim f(x), im f(x), and lim f(x) exist. In case a limit at exists, compute it. 2. у у 1. y = f(x) y = f(x) 1.5 0.5 — X 2 -1 a = 2 a = -1 у у 4. 3. y = f(x) N y = f(x) 1 a=1 a = 0 open Section 2.2 Limits of Functions 101 உmin Namas In Exercises 5-10, by analyzing the graph y = f(x) evaluate f and the limits off at the given point, provided the corresponding limits exist. 2 1 A y = f(x) 1/3 1/6 -5/2 2 4 —2/3 -1 -1 +1/3 DNE 4- 4 ° 0 5. f(-1), lim f(x), limf(x), lim f(x) 6. f(4), lim f(x), lim f(x), lim f(x) 7. FO), lim f(x), lim f(x), lim f(x) DNE 8. f(2), lim f(x), lim f(x), lim fx) pre 9. f(1), lim-f(x), lim f(x), lim f() 0 10. f(-), lim f(x) lim f(x), lim f(x) 1/2 1-2- x1+ 5+ – -212 In Exercises 11-28, evaluate the limits using P1-P6. 11. lim (x – 6×2 – 4) 12. lim (3×4 + 5x + 2x – 1) x-1 13. lim 2x + 3 14. lim VX2 – 3 x→0 15. lim 2x+1 x-+-1 1-3 16. lim -6 -3x+1 17. lim (4 – 2x) 2 18. lim (5 – x2) 102 Chapter 2 Functions, Limits, and Continuity 19. lim -1 * → x2+x+1 20. lim x+1 x → x2-x-1 21. lim x3 5x + 1 22. lim 32×3 x² + 3x – 4 x → X-2 24. lim – 2 x →4 x+1 23. lim ( – ) 25. lim x2 – 4x + 4 26. lim x3 + x – 1 x2 x>6 27. lim x3–2x-3 x→-2 x+3 28. lim x-x4 x → 2x-1 Section 2.3 Continuity 103 5. Elain did the following computation: – -> lim X – 2 X- lim – 2 1 1 lim x 2 x2 – 4 *+2 (x – 2)(x + 2) x+2x + 2 4 Jessica argued that Elain’s computation is incorrect since she canceled x 2 from the numerator and the denominator, and this is not allowed when computing the limit as x → 2. With whom do you agree? 6. Elain looks at the function g:R\{-2,2} R, 8(x) 1 and claims that the functions f and g are not equal. Do you agree with her? Explain. xx² 7. Compute lim and lim x2 -2×2 + x Hint: Simplify first the given fractions. 8. Can you compute lim f(x) using P4? Is the cancellation technique applicable here? Explain. 9. Generate a table of values for x and f(x) with the values of x approaching but never equal to – 2. What do you observe? 10. Conjecture a value for lim f(x), lim f(x), and lim f(x). x→ -2- x-2+ x-2 x? hout generating a table of values for find lim lim x2 x²4² x–2- 24′ x–2+ x2-4′ lim x2 lim and lim Your reasoning should take *0376 +x? -144-2×2+1 X-2 ust another EXERCISES 2.4 In Exercises 1-4, study the continuity off on the real line. -X + 3 1. f(x) = us cholds. inuous on if x = -1 -x² + 2 if x>-1 x – 2 2. f(x) = if x = 2 x – 2×2 – 1 if x > 2 x2 + 2x + 3 if x so x2 – 3x + 1 if x Š 1 3. f(x) = 5x + 3 if x > 0 4. f(x) = 3x 2x + 1 if x > 1 X + 1 In Exercises 5-8, find the values of a for which the function f is continuous on R. 2×2 – 3 if x = 1 x – a if x so 3x – a if x > 1 6. f(x) = x2 + 1 if x > 0 if x = 2 – 2x – az if x = 1 8. f(x) = 1 – 4ax if x > 1 nd (2,00), yzing the 5. f(x) = 7. f(x) = ar? – 6 if x > 2 2. We x→4 Plyno- ine, & In Exercises 9-14, give an example of a function f as specified. 9. f is continuous at x = 4. 10. f is continuous at x = 2. 11. f is discontinuous at x = 4, but lim f(x) exists. 12. lim f(x) exists, and f is discontinuous at x = 2. 13. x4+ 14. lim f(x) = f(2), and f is discontinuous at x = 2. 15. How many continuous functions f: R → R are there that satisfy [f(x)]2 = x² for all x ER? r2 hold, x 4-76) # lim f(x). x2 x-a EXERCISES 2.3 In Exercises 1-4, determine if lim f(x), im f(x), and lim f(x) exist. In case a limit at exists, compute it. 2. у у 1. y = f(x) y = f(x) 1.5 0.5 — X 2 -1 a = 2 a = -1 у у 4. 3. y = f(x) N y = f(x) 1 a=1 a = 0 open Section 2.2 Limits of Functions 101 உmin Namas In Exercises 5-10, by analyzing the graph y = f(x) evaluate f and the limits off at the given point, provided the corresponding limits exist. 2 1 A y = f(x) 1/3 1/6 -5/2 2 4 —2/3 -1 -1 +1/3 DNE 4- 4 ° 0 5. f(-1), lim f(x), limf(x), lim f(x) 6. f(4), lim f(x), lim f(x), lim f(x) 7. FO), lim f(x), lim f(x), lim f(x) DNE 8. f(2), lim f(x), lim f(x), lim fx) pre 9. f(1), lim-f(x), lim f(x), lim f() 0 10. f(-), lim f(x) lim f(x), lim f(x) 1/2 1-2- x1+ 5+ – -212 In Exercises 11-28, evaluate the limits using P1-P6. 11. lim (x – 6×2 – 4) 12. lim (3×4 + 5x + 2x – 1) x-1 13. lim 2x + 3 14. lim VX2 – 3 x→0 15. lim 2x+1 x-+-1 1-3 16. lim -6 -3x+1 17. lim (4 – 2x) 2 18. lim (5 – x2) 102 Chapter 2 Functions, Limits, and Continuity 19. lim -1 * → x2+x+1 20. lim x+1 x → x2-x-1 21. lim x3 5x + 1 22. lim 32×3 x² + 3x – 4 x → X-2 24. lim – 2 x →4 x+1 23. lim ( – ) 25. lim x2 – 4x + 4 26. lim x3 + x – 1 x2 x>6 27. lim x3–2x-3 x→-2 x+3 28. lim x-x4 x → 2x-1 Section 2.3 Continuity 103 5. Elain did the following computation: – -> lim X – 2 X- lim – 2 1 1 lim x 2 x2 – 4 *+2 (x – 2)(x + 2) x+2x + 2 4 Jessica argued that Elain’s computation is incorrect since she canceled x 2 from the numerator and the denominator, and this is not allowed when computing the limit as x → 2. With whom do you agree? 6. Elain looks at the function g:R\{-2,2} R, 8(x) 1 and claims that the functions f and g are not equal. Do you agree with her? Explain. xx² 7. Compute lim and lim x2 -2×2 + x Hint: Simplify first the given fractions. 8. Can you compute lim f(x) using P4? Is the cancellation technique applicable here? Explain. 9. Generate a table of values for x and f(x) with the values of x approaching but never equal to – 2. What do you observe? 10. Conjecture a value for lim f(x), lim f(x), and lim f(x). x→ -2- x-2+ x-2 x? hout generating a table of values for find lim lim x2 x²4² x–2- 24′ x–2+ x2-4′ lim x2 lim and lim Your reasoning should take *0376 +x? -144-2×2+1 X-2 ust another EXERCISES 2.4 In Exercises 1-4, study the continuity off on the real line. -X + 3 1. f(x) = us cholds. inuous on if x = -1 -x² + 2 if x>-1 x – 2 2. f(x) = if x = 2 x – 2×2 – 1 if x > 2 x2 + 2x + 3 if x so x2 – 3x + 1 if x Š 1 3. f(x) = 5x + 3 if x > 0 4. f(x) = 3x 2x + 1 if x > 1 X + 1 In Exercises 5-8, find the values of a for which the function f is continuous on R. 2×2 – 3 if x = 1 x – a if x so 3x – a if x > 1 6. f(x) = x2 + 1 if x > 0 if x = 2 – 2x – az if x = 1 8. f(x) = 1 – 4ax if x > 1 nd (2,00), yzing the 5. f(x) = 7. f(x) = ar? – 6 if x > 2 2. We x→4 Plyno- ine, & In Exercises 9-14, give an example of a function f as specified. 9. f is continuous at x = 4. 10. f is continuous at x = 2. 11. f is discontinuous at x = 4, but lim f(x) exists. 12. lim f(x) exists, and f is discontinuous at x = 2. 13. x4+ 14. lim f(x) = f(2), and f is discontinuous at x = 2. 15. How many continuous functions f: R → R are there that satisfy [f(x)]2 = x² for all x ER? r2 hold, x 4-76) # lim f(x). x2