Find the horizontal asymptote, if any, of the graph of the rational function. (If it exists, it should start with does not exist, type DNE).
[tex]h(x)=\frac{2 a x^3}{5 x^2+1}[/tex]
Answer (2)
To find the horizontal asymptote of h ( x ) = 5 x 2 + 1 2 a x 3 , we examine the limits as x approaches ± ∞ .
Divide the numerator and denominator by x 2 to get lim x → ± ∞ 5 + x 2 1 2 a x .
Evaluate the limits: lim x → ∞ 5 2 a x = ∞ and lim x → − ∞ 5 2 a x = − ∞ .
Since the limits are infinite, the horizontal asymptote does not exist: DNE .
Explanation
Problem Analysis We are asked to find the horizontal asymptote of the rational function h ( x ) = 5 x 2 + 1 2 a x 3 . To do this, we need to examine the behavior of the function as x approaches positive and negative infinity.
Limits to Evaluate To find the horizontal asymptote, we need to compute the limits: x → ∞ lim 5 x 2 + 1 2 a x 3 x → − ∞ lim 5 x 2 + 1 2 a x 3
Dividing by x 2 To evaluate these limits, we divide both the numerator and the denominator by the highest power of x in the denominator, which is x 2 . This gives us: x → ∞ lim x 2 5 x 2 + 1 x 2 2 a x 3 = x → ∞ lim 5 + x 2 1 2 a x x → − ∞ lim x 2 5 x 2 + 1 x 2 2 a x 3 = x → − ∞ lim 5 + x 2 1 2 a x
Evaluating the Limits Now, we evaluate the limits. As x approaches infinity, x 2 1 approaches 0. Thus, we have: x → ∞ lim 5 + x 2 1 2 a x = x → ∞ lim 5 2 a x x → − ∞ lim 5 + x 2 1 2 a x = x → − ∞ lim 5 2 a x Since a is a constant, as x goes to ∞ , 2 a x /5 also goes to ∞ (or − ∞ if a < 0 ). Similarly, as x goes to − ∞ , 2 a x /5 also goes to − ∞ (or ∞ if a < 0 ).
Conclusion Since the limits as x approaches ∞ and − ∞ are infinite, there is no horizontal asymptote.
Final Answer Therefore, the horizontal asymptote does not exist. We denote this as DNE.
Examples
Understanding horizontal asymptotes is crucial in fields like physics and engineering. For instance, when modeling the velocity of an object under constant acceleration but with a drag force, the velocity approaches a horizontal asymptote, representing the terminal velocity. Similarly, in chemical reaction kinetics, the concentration of a reactant might approach a horizontal asymptote as the reaction progresses, indicating the reaction's completion. Recognizing and calculating these asymptotes helps predict long-term behavior and stability in various systems.
To find the horizontal asymptote of the function h ( x ) = 5 x 2 + 1 2 a x 3 , we determine that the degree of the numerator (3) is greater than the degree of the denominator (2), which means there is no horizontal asymptote. Evaluating the limits as x → ∞ and x → − ∞ confirms that the limits approach infinity or negative infinity, leading to the conclusion that the horizontal asymptote does not exist (DNE).
;
