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).


$f\left(x\right)=\frac{1}{2x^4-\sqrt{4x^8-9x^4}}$

Verified step by step guidance

1. Identify the dominant terms in the numerator and the denominator as x approaches infinity and negative infinity. For f(x) = \(\frac{1}{2x^4 - \sqrt{4x^8 - 9x^4}\)}, the dominant term in the denominator is \(\sqrt{4x^8}\) = 2x^4 when x is large.
2. Simplify the expression by factoring out the dominant term from the square root in the denominator: \(\sqrt{4x^8 - 9x^4}\) = x^4\(\sqrt{4 - \frac{9}{x^4}\)}.
3. Rewrite the function f(x) in terms of the dominant term: f(x) = \(\frac{1}{2x^4 - x^4\sqrt{4 - \frac{9}{x^4}\)}} = \(\frac{1}{x^4(2 - \sqrt{4 - \frac{9}{x^4}\)})}.
4. Evaluate the limit as x approaches infinity: lim_{x \(\to\) \(\infty\)} f(x) = lim_{x \(\to\) \(\infty\)} \(\frac{1}{x^4(2 - \sqrt{4 - \frac{9}{x^4}\)})}. As x becomes very large, \(\frac{9}{x^4}\) approaches 0, so \(\sqrt{4 - \frac{9}{x^4}\)} approaches 2, making the denominator approach x^4(2 - 2) = 0. Therefore, the limit is undefined, indicating no horizontal asymptote at infinity.
5. Evaluate the limit as x approaches negative infinity: lim_{x \(\to\) -\(\infty\)} f(x) = lim_{x \(\to\) -\(\infty\)} \(\frac{1}{x^4(2 - \sqrt{4 - \frac{9}{x^4}\)})}. The analysis is similar to the positive infinity case, and the limit is also undefined, indicating no horizontal asymptote at negative infinity.

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 finite number, it indicates that the function approaches a horizontal line at that value.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They provide insight into the end behavior of a function. If a function has a horizontal asymptote at y = L, it means that as x becomes very large or very small, the function values get closer to L, indicating stability in the function's output at extreme values.

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 the polynomials in the numerator and denominator is essential for analyzing limits at infinity and determining horizontal asymptotes. The behavior of these functions can vary significantly based on the relative degrees of P and Q.