Find all horizontal asymptotes of the following function. f(x)=(x-4)/(10x^(2)-56 x+64)

Understand Concept: Understand the concept of horizontal asymptotes for rational functions. Horizontal asymptotes occur where the function f(x) approaches a constant value as x approaches infinity or negative infinity. For rational functions, horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials.Compare Degrees: Compare the degrees of the numerator and the denominator. The degree of the numerator (x-4) is 1, and the degree of the denominator (10x^2 - 56x + 64) is 2.Determine Asymptote: Determine the horizontal asymptote based on the degrees. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.Check Other Asymptotes: Check for any other possible horizontal asymptotes. There are no other possible horizontal asymptotes because the degree of the denominator is strictly greater than the degree of the numerator, which means the function will approach zero as x approaches infinity or negative infinity.

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Find all horizontal asymptotes of the following function.

f(x)=(x-4)/(10x^(2)-56 x+64)

Find all horizontal asymptotes of the following function. $\newline$ $f(x)=\frac{x-4}{10 x^{2}-56 x+64}$

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Q. Find all horizontal asymptotes of the following function.

\newline

f(x)=\frac{x-4}{10 x^{2}-56 x+64}

Understand Concept: Understand the concept of horizontal asymptotes for rational functions. Horizontal asymptotes occur where the function $f(x)$ approaches a constant value as $x$ approaches infinity or negative infinity. For rational functions, horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials.
Compare Degrees: Compare the degrees of the numerator and the denominator. $\newline$ The degree of the numerator $x-4$ is $1$ , and the degree of the denominator $10x^2 - 56x + 64$ is $2$ .
Determine Asymptote: Determine the horizontal asymptote based on the degrees. $\newline$ Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is $y = 0$ .
Check Other Asymptotes: Check for any other possible horizontal asymptotes. There are no other possible horizontal asymptotes because the degree of the denominator is strictly greater than the degree of the numerator, which means the function will approach zero as $x$ approaches $\infty$ or $-\infty$ .