Domanda del libro di testo
Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
5 ―x²
lim ------------- = 0
x → ―2 (√g(x))
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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 2.5.4
In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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Guida verificata passo dopo passo1
Step 1: Understand the definition of continuity. A function is continuous on an interval if it is continuous at every point in that interval. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
Step 2: Identify the interval of interest, which is [-1, 3]. This means we need to check the continuity of the function at every point within this interval, including the endpoints -1 and 3.
Step 3: Examine the graph of the function over the interval [-1, 3]. Look for any breaks, jumps, or holes in the graph, as these indicate points where the function may not be continuous.
Step 4: Check the endpoints of the interval. Ensure that the function is defined at x = -1 and x = 3, and that the limits from the left and right at these points match the function's value at these points.
Step 5: Identify any points within the interval where the function is not continuous. This could be due to a jump discontinuity, an infinite discontinuity, or a removable discontinuity. Note these points and explain why the function fails to be continuous at each one.
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Continuity of Functions
A function is continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. Formally, a function f(x) is continuous at a point c if the limit of f(x) as x approaches c equals f(c). Understanding this concept is crucial for determining whether a function is continuous over a specified range.
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Intro to Continuity
Types of Discontinuities
Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by redefining it. A jump discontinuity happens when the left-hand and right-hand limits at a point do not match, while an infinite discontinuity occurs when the function approaches infinity at a point. Identifying these types helps in pinpointing where a function fails to be continuous.
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05:25Determine Continuity Algebraically
Evaluating Limits
Limits are fundamental in analyzing the behavior of functions as they approach specific points. To determine continuity, one must evaluate the limit of the function at the endpoints and any critical points within the interval. If the limit exists and equals the function's value at those points, the function is continuous there. This concept is essential for assessing continuity over a given interval.
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05:50One-Sided Limits
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If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.
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Domanda del libro di testo
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
llimx→0 (x −x cos x) / sin² 3x
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Using the Formal Definition
Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.
f(x) = −3x − 2, c = −1, ε = 0.03
299
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Domanda del libro di testo
Horizontal and Vertical Asymptotes
Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .
266
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Domanda del libro di testo
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x→∞ √(x² + 1) / (x + 1)
269
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