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Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 3x − 1 = 0
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Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 2, Problema 2.6.93
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (−1 / x²) = −∞
Guida verificata passo dopo passo1
Understand the formal definition of a limit approaching negative infinity: For every positive number M, there exists a positive number δ such that for all x, 0 < |x| < δ implies f(x) < -M.
Identify the function f(x) = -1/x² and the limit statement lim x → 0 (−1 / x²) = −∞.
Given any positive number M, we need to find a δ > 0 such that for all x, 0 < |x| < δ, the inequality -1/x² < -M holds.
Rearrange the inequality -1/x² < -M to 1/x² > M, which implies x² < 1/M. Therefore, |x| < 1/√M.
Choose δ = 1/√M. Then, for all x such that 0 < |x| < δ, it follows that 1/x² > M, which satisfies the condition for the limit to be -∞.
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Limit Definition
The formal definition of a limit involves showing that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. In this case, the limit is approaching negative infinity, which requires a modified approach to demonstrate that f(x) becomes arbitrarily large negative as x approaches 0.
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Percorso guidato
05:43
Definition of the Definite Integral
Behavior of Rational Functions
Understanding the behavior of rational functions, particularly as x approaches a value where the denominator tends to zero, is crucial. For the function −1/x², as x approaches 0, the denominator becomes very small, causing the function value to grow negatively without bound, leading to a limit of negative infinity.
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6:04Intro to Rational Functions
Infinity in Limits
When dealing with limits that approach infinity, the concept of infinity in calculus is used to describe unbounded behavior. A limit approaching negative infinity means the function values decrease without bound as x approaches the specified point. This requires demonstrating that for any large negative number, the function can be made smaller than that number by choosing x sufficiently close to the limit point.
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03:07Cases Where Limits Do Not Exist
Pratica correlata
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Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = (√(x² + 4)) / x
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Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0 (−1) / (x² (x + 1))
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Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = −3/(x − 3)
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Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=x³−3x²+4, P(2,0)
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Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.
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