Determine the end behavior of each function as x →+∞and as x...

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Key Concepts

End Behavior

The end behavior of a function describes what happens to the outputs (y values) as the inputs (x values) become extremely large or extremely small. It involves analyzing limits as x approaches positive infinity and negative infinity, which can indicate the existence of horizontal asymptotes or other long-term trends.

Negative Exponents

Functions with negative exponents, such as x??, can be rewritten as the reciprocal of a positive power (1/x? in this case). This transformation is key to understanding the behavior of the function at the extremes of the domain, since the reciprocal function diminishes towards zero as the absolute value of x increases.

Rational Functions

When a function is expressed as a constant multiplied by a negative power of x, it takes the form of a rational function where the numerator is a constant and the denominator is a power of x. The analysis of such functions involves evaluating limits by considering the dominant effect of the denominator as x becomes very large or very small, often resulting in horizontal asymptotes.

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