Domanda del libro di testo
Evaluate each limit and justify your answer.
lim x→4 √x^3−2x^2−8x / x−4
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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 2.4
Determine the following limits at infinity.
lim x→−∞ 3x^11
Guida verificata passo dopo passo1
Identify the function whose limit we need to find: \( f(x) = 3x^{11} \).
Recognize that as \( x \to -\infty \), the term \( x^{11} \) will dominate the behavior of the function because it is a polynomial.
Since the exponent 11 is odd, \( x^{11} \) will be negative when \( x \) is negative.
Multiply by the constant 3, which will not change the sign of the expression, so \( 3x^{11} \) will also be negative as \( x \to -\infty \).
Conclude that the limit of \( 3x^{11} \) as \( x \to -\infty \) is \(-\infty\).
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Limits at Infinity
Limits at infinity refer to the behavior of a function as the input approaches positive or negative infinity. This concept helps in understanding how functions behave for very large or very small values of x, which is crucial for analyzing polynomial, rational, and other types of functions.
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Polynomial Functions
Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The degree of the polynomial, which is the highest power of the variable, significantly influences its end behavior as x approaches infinity or negative infinity.
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End Behavior of Functions
End behavior describes how a function behaves as the input values become very large or very small. For polynomial functions, the leading term (the term with the highest degree) dominates the function's behavior at infinity, determining whether the limit approaches positive or negative infinity.
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