(a) find the vertical asymptotes of the graph of f(x) . (b) Describe the behavior of f(x) to the

step 1 For this problem, we are asked to find the vertical asymptotes of the function f of x equals x s

(a) find the vertical asymptotes of the graph of f(x) . (b)...

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Key Concepts

Vertical Asymptote

A vertical asymptote is a line x = a where a rational function becomes unbounded as x approaches a. It typically occurs when the denominator of the function is zero while the numerator is nonzero, indicating that the function will approach positive or negative infinity near that point.

Limit Analysis Near Vertical Asymptotes

This concept involves studying the behavior of a function as it approaches its vertical asymptote from the left and right. By evaluating the one-sided limits, one can determine whether the function tends toward positive or negative infinity on either side of the asymptote, which helps in sketching the graph of the function.

Rational Functions

Rational functions are ratios of two polynomials. Understanding them involves determining their domain, identifying points of discontinuity, and analyzing their asymptotic behavior. Techniques such as factoring and canceling common factors are often used to simplify the function and reveal hidden characteristics like vertical asymptotes.

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