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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = X L2
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. +1 w(t) = P-1
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = COS 1 1 X X # 0 x = 0
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. x-2 f(x) = |x-21 -1 X #2 x
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.ƒ(x) = 3x2/3 − 9x3
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.g(t) = 3t−2/3 − 9t3
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = 2x²-50 x+5
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.ƒ(x) = tan(sin x)
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.g(t) = tan 2t
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous.ƒ(x) = csc(x2)
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = 2[x/2] + 4[x/4]
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. f(x)=√x² +9
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = (x3 + 3)5/2
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = cos([x])
Determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left- or right-continuous. f(x) = [x+3] + [2x]
The graph of the following function is shown in Figure 18.Show that f is continuous for x ≠ 1, 2. Then compute the right- and left-hand limits at x = 1, 2, and determine whether ƒ is
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. f(x) = x² x + x1/4
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = x1/3 + x3/4
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = x2 − 3x1/2
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = x−4/3
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = tan2 x
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = cos3 x
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = cos(x1/3 + 1)
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. f(x) = tan³ (x - 2) 9x² + 2
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.ƒ(x) = (x4 + 1)3/2
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. f(x) = cos(x²) x² - 1
Draw the graph of ƒ(x) = x − [x]. At which points is ƒ discontinuous? Is it left-or right continuous at those points?
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim h→0 √2+h-2 h
Evaluate using the Squeeze Theorem. lim(2¹ - 1) cos 1-0 1 t
Evaluate using the Squeeze Theorem. x-3 |x - 3| lim(x²-9), x→3
Evaluate using the Squeeze Theorem. 1 - X lim x² cos- x→0
Evaluate using the Squeeze Theorem. lim(x - 1) sin Л Л x-1
Evaluate using the Squeeze Theorem. lim x sin x-0 1 x² ic
Evaluate using the Squeeze Theorem. lim √x3cos(x/x) X→0+
What does the Squeeze Theorem say about if the limits and ƒ, u, and l are related as in Figure 8? The inequality ƒ(x) ≤ u(x) is not satisfied for all x. Does this affect the validity of your
Evaluate using the Squeeze Theorem. lim (3¹ - 1) sin 1-0-
Evaluate using the Squeeze Theorem. lim(t²-4) cos 1-2 1 t-2
State precisely the hypothesis and conclusions of the Squeeze Theorem for the situation in Figure 6. 2. 1 FIGURE 6 y = u(x) y=f(x) y=1(x) -X 2
Evaluate using the Squeeze Theorem. lim tan x cos sin X-0 X
Evaluate using the Squeeze Theorem. lim cos cos(tan 8) 0→
In Figure 7, is ƒ squeezed by u and l at x = 3? At x = 2? 1.5+ 1 2 3 4 y= u(x) y=f(x) y=1(x) -X
Plot the graphs of on the same set of axes. What can you say about if f is squeezed by l and u at x uis = (x)/ pur | / - x + 1 = (x)n
State whether the inequality provides sufficient information to determine and if so, find the limit. lim f(x), x-1
Determine assuming that cos x ≤ ƒ(x) ≤ 1. lim f(x) .0←x
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
In Exercises 17–26, evaluate using Theorem 2 as necessary. THEOREM 2 Important Trigonometric Limits sin 0 lim 0-0 0 = 1 and lim 0-0 1- cos 0 0 = 0 =
Evaluate using a substitution θ = 11x. lim X-0 sin 11x X
Evaluate Multiply the numerator and denominator by (7)(11)t. lim sin 7t 10 Sin 11t
In Exercises 29–48, evaluate the limit. sin 4h lim h→0 4h
In Exercises 29–48, evaluate the limit. sin 9h lim h→0 h
In Exercises 29–48, evaluate the limit. X lim x→ sin 3x
In Exercises 29–48, evaluate the limit. sin h lim h→0 5h
In Exercises 29–48, evaluate the limit. sin 70 lim 0-0 sin 30
In Exercises 29–48, evaluate the limit. tan 4t lim 1-0 t sect
In Exercises 29–48, evaluate the limit. tan 4x lim x-0 tan 9x
In Exercises 29–48, evaluate the limit. tan 4x lim x-0 9x
In Exercises 29–48, evaluate the limit. lim sin 2h sin 3h h²
In Exercises 29–48, evaluate the limit. sin(z/3) lim 2-0 sin z
In Exercises 29–48, evaluate the limit. csc 8t lim 1-0 csc 4t CSC
In Exercises 29–48, evaluate the limit. sin(-30) lim 0-0 sin 40
Use the identity sin 2θ = 2 sinθ cos θ to evaluate lim 0→0 sin 20 - 2 sin 0 0²
In Exercises 29–48, evaluate the limit. lim x-0 sin 3x sin 2x x sin 5x
In Exercises 29–48, evaluate the limit. sin 5x sin 2x lim x→0 sin 3x sin 5x
In Exercises 29–48, evaluate the limit. lim h→0 1 - cos 2h h
In Exercises 29–48, evaluate the limit. lim h→0 sin(2h)(1- cos h) h²
In Exercises 29–48, evaluate the limit. lim 0-0 cos 20 - cos 0 0
In Exercises 29–48, evaluate the limit. 1 - cos 2t lim 1-0 sin² 3t
In Exercises 29–48, evaluate the limit. lim h→ { 1 - cos 3h h
Use the identity sin 3θ = 3 sinθ − 4 sin3 θ to evaluate lim 0→0 sin 303 sin 0 03
Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity. lim x² + 2x + 1 x + 1
Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity. lim X-6 x² - 36 X-6
Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity. 9-h² lim h-3 h - 3
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). x-7 lim x-7 x² - 49
Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity. 2t - 18 lim 1-9 5t - 45
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim x→-2 x² + 3x + 2 x + 2
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). X x² - 64 9 lim X-8 x
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim x→8 x³ - 64x x-8
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim h→0 (1 + h)³ - 1 h
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim 2x + 1 2x² + 3x + 1
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim x-5 2x²-9x-5 x² - 25
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim h→4 (h+ 2)² - 9h h - 4
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). - zt - X lim x+3x²9
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim X-2 3x² - 4x-4 2x² - 8
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim h→0 (3 + h)³ - 27 h
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim x 16 √x-4 x 16
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). 42-1 lim 1-0 4-1
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim h→0 1 (h+ 2)² h 1 4
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). 2t+4 lim 1-2 12 - 31²
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). √x-4-2 lim x→8 x-8
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). x - 4 lim x-4 √√x - √8-x x→4
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). y² + y - 12 lim y-3 y3 - 10y + 3 y→3
Evaluate the limit, if it exists. If not, determine whether the one-sided limits exist (finite or infinite). lim X-4 √5-x-1 2-√√x

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