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Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim θ→0 (2 − cot θ)
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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 50
Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.
lim x→(−π/2)⁺ sec x
Guida verificata passo dopo passo1
Understand the function: The secant function, sec(x), is defined as 1/cos(x). As x approaches a certain value, we need to consider the behavior of the cosine function.
Identify the point of interest: We are looking at the limit as x approaches -π/2 from the right (denoted as x→(-π/2)⁺). This means we are considering values of x that are slightly greater than -π/2.
Analyze the cosine function: At x = -π/2, cos(x) is 0. As x approaches -π/2 from the right, cos(x) approaches 0 from the positive side, meaning cos(x) is positive but very small.
Consider the behavior of sec(x): Since sec(x) = 1/cos(x), as cos(x) approaches 0 from the positive side, sec(x) will become very large. This indicates that sec(x) approaches positive infinity.
Conclude the limit: Based on the analysis, the limit of sec(x) as x approaches -π/2 from the right is ∞.
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Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points of discontinuity or infinity. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.
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05:50
One-Sided Limits
Secant Function
The secant function, denoted as sec(x), is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). It is important to understand the behavior of sec(x) as x approaches certain values, particularly where cos(x) equals zero, since this leads to undefined values and vertical asymptotes in the secant function.
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Percorso guidato
6:22Graphs of Secant and Cosecant Functions
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (−) or the right (+). In the context of the given limit, evaluating the right-hand limit as x approaches −π/2 helps determine the behavior of sec(x) near this critical point, which is essential for understanding discontinuities in the function.
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05:50One-Sided Limits
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Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.
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Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.
lim x/(x² − 1) as
d. x→−1⁻
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[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
c. It can be shown that the exact value of the solution in part (b) is
(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³
Evaluate this exact answer and compare it with the value you found in part (b).
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[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?
g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2
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