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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.4
Chapter 2, Problem 2.4

Determine the following limits at infinity.


lim x→−∞ 3x^11

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Related Practice
Textbook Question

Use the precise definition of infinite limits to prove the following limits.


limx→−11(x+1)4=∞{\(\displaystyle\]\lim\)_{x\(\to\)-1}}\(\frac{1}{\left(x+1\right)^4}\)=\(\infty\)

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 (8x+5)=13

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→5 1/x^2=1/25

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Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=12x4−4x8−9x4f\(\left\)(x\(\right\))=\(\frac{1}{2x^4-\sqrt{4x^8-9x^4}\)}

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→−3 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)