Find the limit or show that it does not exist. limx →∞(1 - x^2)/(x^3 - x + 1)

Name: Problem 16, Chapter 2: Calculus: Early Transcendentals
Uploaded: 2021-08-26T00:49:40-08:00
Duration: 1 min 37 s
Channel: Anthony Han
Description: Find the limit or show that it does not exist. limx →∞(1 - x^2)/(x^3 - x + 1)

step 1 So here we're given information about a specific limit. We have the limit as X approaches infini

Find the limit or show that it does not exist. limx →∞(1 -...

Key Concepts

Limits at Infinity

This concept refers to the behavior of functions as the independent variable grows without bound. It is used to analyze the end behavior of functions and determine steady-state values or tendencies, which can be finite numbers, infinity, or non-existence of a limit.

Rational Functions

Rational functions are functions expressed as the ratio of two polynomials. Their behavior, especially for large inputs, is largely determined by the degrees and coefficients of the leading terms of the numerator and denominator.

Degree Comparison

For rational functions, comparing the degrees of the numerator and denominator simplifies the determination of limits at infinity. If the degree of the numerator is less than that of the denominator, the limit is zero; if the degrees are equal, the limit is the ratio of the leading coefficients; and if the numerator’s degree is greater, the limit is typically infinite or does not exist.

Dominant Terms

In the context of limits at infinity, the dominant terms are the highest power terms in the numerator and denominator. They overwhelmingly determine the function's behavior when the variable approaches infinity, allowing lower degree terms to be ignored for simplification.

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