15-38 Find the limit or show that it does not exist. limx →∞ ((2 x^2+1)^2)/((x-1)^2(x^2+x))

Name: Problem 21, Chapter 2: Calculus Early Transcendentals
Uploaded: 2020-10-19T15:41:42-08:00
Duration: 1 min 13 s
Channel: Stephen Hobbs
Description: 15-38 Find the limit or show that it does not exist. limx →∞ ((2 x^2+1)^2)/((x-1)^2(x^2+x))

step 1 To evaluate this limit as our variable approaches infinity, we're going to use the ratio. Well,

15-38 Find the limit or show that it does not exist. limx →∞...

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Key Concepts

Limit at Infinity

This concept involves evaluating the behavior of a function as the variable approaches infinity. In the context of rational functions, it typically involves analyzing the dominant terms of the numerator and denominator to determine the end behavior of the function.

Rational Functions

Rational functions are functions that are the ratio of two polynomials. Their limits at infinity can often be determined by comparing the degrees of the numerator and denominator. If these degrees are equal, the limit is the ratio of the leading coefficients; if the degree of the numerator is less, the limit is zero; and if it is greater, the limit does not exist in the conventional sense (it diverges).

Leading Term Analysis

This technique simplifies finding limits by focusing on the highest degree terms in the numerator and denominator, since these dominate the behavior of the function as the variable grows large. By analyzing these terms, one can efficiently determine the asymptotic behavior without needing to consider lower-order terms.

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