Find the limit of each rational function (a) as x →∞and (b) as x →-∞. g(x)=(1)/(x^3-4 x+1)

Name: Problem 18, Chapter 2: Calculus
Uploaded: 2020-08-05T14:26:26-08:00
Duration: 43 s
Channel: Subhadeepta Sahoo
Description: Find the limit of each rational function (a) as x →∞and (b) as x →-∞. g(x)=(1)/(x^3-4 x+1)

step 1 Now in this question for both parts of A and B we can write limit of X tends to plus or minus in

Find the limit of each rational function (a) as x →∞and (b)...

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Key Concepts

Rational Functions

A rational function is a function defined by the ratio of two polynomials. The behavior of rational functions is often determined by the degrees of the numerator and the denominator, which is crucial for analyzing their limits and asymptotic behavior as x approaches infinity or negative infinity.

Limits at Infinity

Limits at infinity involve studying the behavior of a function as the variable grows very large in magnitude. For rational functions, this often means determining whether the function approaches a finite value (horizontal asymptote), grows without bound, or approaches zero, depending on the degrees of the polynomials in the numerator and denominator.

Dominant Term Analysis

Dominant term analysis is the process of identifying the term in a polynomial that has the greatest impact on the function's behavior as the variable becomes very large or very small. In the context of rational functions, comparing the highest power terms in the numerator and denominator allows one to simplify the limit calculation by focusing on these dominant terms.

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