In Exercises 47-56, find the limit of each rational function (a) as x →∞and (b) as x →-∞. h(x)=(

step 1 In this case we have to find the following limit so we have limit x is tending towards plus or m

In Exercises 47-56, find the limit of each rational function...

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches as the variable goes to infinity or negative infinity. For rational functions, when the degrees of the numerator and the denominator are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. This concept helps in determining the long-term behavior of the function.

Rational Functions

Rational functions are expressions defined as the ratio of two polynomials. Understanding their structure is important for analyzing their behavior, particularly as the variable approaches infinity or negative infinity. The properties of rational functions, such as discontinuities, asymptotic behavior, and intercepts, depend on the degrees and coefficients of the numerator and denominator.

Dominant Term Analysis

Dominant term analysis involves identifying the terms in the numerator and denominator with the highest degrees since these terms dictate the behavior of the function as the variable grows large in magnitude. This approach simplifies the process of finding limits at infinity, as lower degree terms become negligible compared to the dominant ones.

End Behavior of Functions

The end behavior of functions describes how functions behave as the input values approach positive or negative infinity. For rational functions, this is typically determined by comparing the degrees of the polynomials in the numerator and denominator. A clear understanding of end behavior is crucial for sketching graphs and understanding limits at infinity.

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