Find the limit of each rational function (a) as x →∞and (b) as x →-∞. Write ∞or -∞where appropri

step 1 Given the equation 3x plus 7 divide by x square minus 2, we have to find out its limit when x ap

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

Key Concepts

Rational Functions

A rational function is a quotient of two polynomials. Understanding rational functions involves analyzing the behavior of both the numerator and denominator, particularly as the variable approaches extreme values (like infinity). This concept is fundamental in algebra and calculus for understanding function behavior and asymptotic properties.

Degree Comparison in Polynomials

When dealing with limits at infinity for rational functions, comparing the degrees of the numerator and the denominator is crucial. The relative degree determines whether the function approaches zero, infinity, or a finite non-zero limit. If the degree of the numerator is less than the degree of the denominator, the limit is zero; if they are equal, the limit is the ratio of the leading coefficients; and if the numerator's degree is greater, the limit typically tends to infinity or negative infinity.

Dominant Term Analysis

Dominant term analysis involves isolating the highest degree (largest power) terms in both the numerator and the denominator. These terms have the greatest impact on the function's behavior for very large absolute values of the variable. This method simplifies the process of finding limits at infinity and understanding end behavior.

Limits at Infinity

Limits at infinity examine the behavior of functions as the independent variable grows without bound, either positively or negatively. In the context of rational functions, determining these limits involves identifying which parts of the function dominate as x becomes very large or very small, thereby establishing the function’s end behavior.

Horizontal Asymptotes

Horizontal asymptotes represent constant values that a function approaches as the independent variable tends to positive or negative infinity. In rational functions, the existence and value of horizontal asymptotes are determined by the degrees of the numerator and denominator. These asymptotes provide insight into the long-term behavior of the function.

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