15-38 Find the limit or show that it does not exist. limx →-∞ (1+x^6)/(x^4+1)

Name: Problem 32, Chapter 2: Calculus Early Transcendentals
Uploaded: 2020-10-19T16:24:59-08:00
Duration: 1 min 28 s
Channel: Stephen Hobbs
Description: 15-38 Find the limit or show that it does not exist. limx →-∞ (1+x^6)/(x^4+1)

step 1 The limit, as X approaches, negative infinity, will look to the highest power of X in the numera

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

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Key Concepts

Comparison of Polynomial Degrees

This concept entails comparing the highest degrees of the polynomials in the numerator and the denominator of a rational function. The degrees determine the limit behavior: if the numerator's degree is higher than the denominator's, the limit tends to infinity or negative infinity; if they are equal, the limit is the ratio of the coefficients of the highest degree terms; and if the numerator's degree is lower, the limit is zero.

Dominant Term Analysis

Dominant term analysis is based on the observation that in a polynomial, the term with the highest degree grows significantly faster than the other terms as the variable becomes very large. In the context of limits at infinity, isolating these dominant terms allows one to simplify the expression and determine the asymptotic behavior of the rational function.

Rational Function Limits at Infinity

This concept involves evaluating the behavior of a function that is expressed as a ratio of two polynomials as the variable approaches infinity or negative infinity. The key idea is that, for large absolute values of the variable, the highest power terms in the numerator and denominator determine the value of the limit, making the evaluation process more straightforward.

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