Finding vertical asymptotes Find all vertical asymptotes x=...

Finding vertical asymptotes Find all vertical asymptotes $x=$ a of the follow ing functions. For each value of $a$, determine $\lim _{x \rightarrow a^{+}} f(x), \lim _{x \rightarrow a^{-}} f(x),$ and $\lim _{x \rightarrow a} f(x)$
$$f(x)=\frac{x^{2}-9 x+14}{x^{2}-5 x+6}$$

Key Concepts

Limits

Limits describe the behavior of a function as the input approaches a specific value. When analyzing rational functions, limits are used to determine how the function behaves near points where it is not defined, such as near vertical asymptotes or removable discontinuities.

Vertical Asymptotes

A vertical asymptote is a vertical line x = a where a function's value increases or decreases without bound as x approaches a. In rational functions, vertical asymptotes are typically found where the denominator equals zero while the numerator is non-zero, indicating a non-removable discontinuity where the function diverges.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a given value from one direction—either from the right (x ? a?) or from the left (x ? a?). These limits are essential when analyzing vertical asymptotes because the behavior of the function may differ on either side of the asymptotic value.

Rational Functions

Rational functions are defined as the quotient of two polynomial functions. Their properties, including potential discontinuities like vertical asymptotes or removable discontinuities, depend on both the numerator and the denominator, especially when the denominator can become zero.

Factorization

Factorization involves rewriting a polynomial as a product of its factors. This technique is crucial in simplifying rational functions to identify common factors between the numerator and denominator. Such simplification can reveal removable discontinuities and help locate vertical asymptotes by highlighting values of x that make the denominator zero.

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