Domanda del libro di testo
Which of the following functions are continuous for all values in their domain? Justify your answers.
c. T(t)=temperature t minutes after midnight in Chicago on January 1
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 1
Explain the meaning of lim x→−∞ f(x)=10.
Guida verificata passo dopo passo1
Step 1: Understand the notation: The expression \( \lim_{x \to -\infty} f(x) = 10 \) is read as 'the limit of \( f(x) \) as \( x \) approaches negative infinity is 10.'
Step 2: Conceptualize the behavior: This means that as \( x \) becomes very large in the negative direction (i.e., \( x \) goes to negative infinity), the values of the function \( f(x) \) get closer and closer to 10.
Step 3: Visualize the graph: Imagine the graph of \( f(x) \). As you move left along the x-axis towards negative infinity, the y-values (outputs of \( f(x) \)) approach the horizontal line \( y = 10 \).
Step 4: Consider the horizontal asymptote: The line \( y = 10 \) can be considered a horizontal asymptote of the function \( f(x) \) as \( x \to -\infty \). This means the graph of \( f(x) \) gets closer to this line but may not necessarily touch or cross it.
Step 5: Relate to real-world scenarios: In practical terms, this limit could represent a situation where a quantity stabilizes at a certain value (10 in this case) as time or another variable decreases without bound.
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Limit at Infinity
The limit at infinity describes the behavior of a function as the input approaches infinity or negative infinity. Specifically, lim x→−∞ f(x)=10 indicates that as x decreases without bound, the values of f(x) approach 10. This concept is crucial for understanding how functions behave in extreme cases.
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One-Sided Limits
Horizontal Asymptote
A horizontal asymptote is a horizontal line that a graph approaches as x approaches infinity or negative infinity. In this case, the line y=10 serves as a horizontal asymptote for the function f(x) as x approaches negative infinity, indicating that f(x) stabilizes around this value rather than diverging.
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Function Behavior
Understanding function behavior involves analyzing how a function changes as its input varies. The statement lim x→−∞ f(x)=10 suggests that for very large negative values of x, the function f(x) behaves consistently, converging to the value 10, which is essential for predicting the function's long-term trends.
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5:46Graphs of Exponential Functions
Pratica correlata
Domanda del libro di testo
Which of the following functions are continuous for all values in their domain? Justify your answers.
b. n(t)=number of quarters needed to park legally in a metered parking space for t minutes
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Domanda del libro di testo
Which of the following functions are continuous for all values in their domain? Justify your answers.
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