Find all vertical asymptotes x=a of the following functions....

Find all vertical asymptotes $x=$a of the following functions. For each value of a, discuss $\lim _{x \rightarrow a^{+}} f(x), \lim _{x \rightarrow a^{-}} f(x),$ and $\lim _{x \rightarrow a} f(x)$ $$f(x)=\frac{x^{3}-10 x^{2}+16 x}{x^{2}-8 x}$$

Key Concepts

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a specific value from the left (x ? a?) or the right (x ? a?). This concept is vital when analyzing functions with vertical asymptotes because the function’s behavior can differ significantly on either side of the asymptote, indicating whether the function tends towards positive or negative infinity.

Removable Discontinuities (Holes)

A removable discontinuity, often referred to as a hole, occurs when a factor in the denominator that causes division by zero cancels with an identical factor in the numerator. Although the function is undefined at that exact point, the limit exists and the behavior of the function near the point is typically well-behaved compared to vertical asymptotes.

Rational Functions

A rational function is the ratio of two polynomials. Their behavior is largely determined by the zeros of the denominator and how they interact with the numerator. Understanding the structure of rational functions is important for identifying discontinuities, such as vertical asymptotes, and other features like holes and end behavior.

Vertical Asymptotes

Vertical asymptotes occur at the values of x where the denominator of a rational function is zero and the function tends toward infinity from either side, indicating a nonremovable discontinuity. They are found by setting the denominator equal to zero and checking that these zeros do not cancel with similar factors in the numerator.

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