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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 2 - Limits and ContinuityProblema 1j
Capitolo 2, Problema 1j

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?


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j. limx→2− f(x) = 2

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To determine the limit of the function as x approaches 2 from the left, we need to analyze the graph of the function y = f(x) near x = 2.
Look at the values of f(x) as x approaches 2 from values less than 2 (i.e., from the left side). This is denoted as limx→2− f(x).
Observe the behavior of the graph as x gets closer to 2 from the left. Check if the y-values are approaching a specific number.
If the y-values are approaching 2 as x approaches 2 from the left, then the statement limx→2− f(x) = 2 is true.
If the y-values are approaching a number other than 2, or if the graph does not approach a specific value, then the statement is false.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, even if they are not defined at those points. For example, the limit of f(x) as x approaches 2 from the left (denoted as lim x→2− f(x)) examines the values of f(x) as x gets closer to 2 from values less than 2.
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One-Sided Limits

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only. The left-hand limit (lim x→c− f(x)) considers values approaching c from the left, while the right-hand limit (lim x→c+ f(x)) considers values approaching c from the right. Understanding one-sided limits is crucial for analyzing functions that may have different behaviors from each side of a point.
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One-Sided Limits

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the graph of a function to determine the value that the function approaches as the input approaches a certain point. By observing the behavior of the graph near that point, one can infer the limit. This visual approach is particularly useful for identifying discontinuities or jumps in the function, which can affect the limit's value.
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Finding Limits Numerically and Graphically
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Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



c. limx→0− f(x) = 0

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Domanda del libro di testo

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



f. limx→0 f(x) = 0

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Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limx→0 (tan 2x) / x

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Limits of quotients


Find the limits in Exercises 23–42.


limx→−3 (2 − √(x² − 5)) / (x + 3)

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Domanda del libro di testo

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?


e. limx→0 f(x) exists

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Domanda del libro di testo

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

         

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

         

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

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