Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).


$f\left(x\right)=\frac{6x^2-9x+8}{3x^2+2}$

Verified step by step guidance

1. Identify the degrees of the polynomials in the numerator and the denominator. Here, both the numerator and the denominator are quadratic polynomials with the highest degree term being x^2.
2. For limits at infinity, focus on the leading terms of the numerator and the denominator. The leading term in the numerator is 6x^2 and in the denominator is 3x^2.
3. Calculate the limit as x approaches infinity by dividing the leading coefficients: lim_{x→∞} f(x) = lim_{x→∞} (6x^2/3x^2) = 6/3 = 2.
4. Similarly, calculate the limit as x approaches negative infinity. Since the leading terms are the same, the limit will also be the same: lim_{x→-∞} f(x) = 2.
5. Since both limits as x approaches positive and negative infinity are equal to 2, the horizontal asymptote of the function f(x) is y = 2.

Key Concepts

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity is a constant, it indicates that the function approaches that constant value, suggesting a horizontal asymptote.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They provide insight into the long-term behavior of a function. For rational functions, horizontal asymptotes can often be found by comparing the degrees of the numerator and denominator, which helps in predicting the function's end behavior.

Rational Functions

Rational functions are ratios of polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the degrees of these polynomials is essential for analyzing limits and asymptotic behavior. In the given function, the degrees of the numerator and denominator will determine the horizontal asymptotes and the limits at infinity.