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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 2.36
Chapter 2, Problem 2.36

Determine the following limits.
lim x→∞ (x4 − 1) / (x5 + 2)

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1
Identify the highest power of x in both the numerator and the denominator. Here, the highest power in the numerator is x^4 and in the denominator is x^5.
Divide every term in the numerator and the denominator by x^5, the highest power of x in the denominator.
Rewrite the expression: (x^4/x^5 - 1/x^5) / (x^5/x^5 + 2/x^5).
Simplify the expression: (1/x - 1/x^5) / (1 + 2/x^5).
Evaluate the limit as x approaches infinity. As x approaches infinity, terms with x in the denominator approach zero, simplifying the expression to 0/1.

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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 involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, which can help determine horizontal asymptotes and the end behavior of polynomial functions.
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One-Sided Limits

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit, both the numerator and denominator are polynomials, and their degrees play a significant role in determining the limit's value as x approaches infinity.
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Introduction to Polynomial Functions

Dominant Term

The dominant term in a polynomial is the term with the highest degree, which significantly influences the function's behavior as x approaches infinity. In the limit provided, identifying the dominant terms in both the numerator and denominator allows for simplification and accurate evaluation of the limit.
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Simplifying Trig Expressions Example 1
Related Practice
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\(\right\))=\(\frac{1}{2x^4-\sqrt{4x^8-9x^4}\)}

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Textbook Question

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

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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→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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Textbook Question

If a function f represents a system that varies in time, the existence of lim limt→∞f(t){\(\displaystyle\[\lim\)_{t\(\rightarrow\]\infty\)}{f(t)}} means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.


The population of a culture of tumor cells is given by p(t)=3500tt+1p\(\left\)(t\(\right\))=\(\frac{3500t}{t+1}\).

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Textbook Question

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

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Textbook Question

Determine the following limits.


limx→0x3−5x2x2{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\frac{x^3-5x^2}{x^2}\)

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