For function find all asymptotes and the coordinates of any holes in its graph. f(x)=(x^2-2 x+3)

step 1 19th so here we have effects as x square minus 2x plus 3 over x plus 3 all right so one thing is

For function find all asymptotes and the coordinates of any...

Key Concepts

Rational Functions

A rational function is defined as the quotient of two polynomials. Its analysis involves understanding the behavior of its numerator and denominator, especially where the function may not be defined or may approach infinity.

Polynomial Division

Polynomial division is used to simplify rational functions, especially when the degree of the numerator is greater than or equal to the degree of the denominator. This process can reveal oblique (or slant) asymptotes by expressing the function as a polynomial plus a remainder over the original denominator.

Vertical Asymptote

Vertical asymptotes occur where the denominator of a rational function is equal to zero and is not canceled by the numerator. These lines in the graph represent values of the variable where the function grows without bound.

Oblique Asymptote

An oblique asymptote is a diagonal line that the graph of a rational function approaches as the variable tends to positive or negative infinity. This occurs when the degree of the numerator is exactly one more than that of the denominator, and the asymptote can be found through polynomial division.

Removable Discontinuity

A removable discontinuity, often referred to as a 'hole', occurs when a factor in the numerator and denominator cancels out, leaving the function undefined at that specific point. Although the function is not defined at the hole, its limit exists there, indicating a gap in the graph rather than an asymptote.

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