Consider the following equation:
[tex]f(x)=\frac{x^2+4}{4 x^2-4 x-8}[/tex]
Name the vertical asymptote(s).
Name the horizontal asymptote(s).
Answer (1)
Find the zeros of the denominator 4 x 2 − 4 x − 8 = 0 , which gives vertical asymptotes x = − 1 and x = 2 .
Check that the numerator is non-zero at these points.
Evaluate the limit of the function as x approaches infinity to find the horizontal asymptote: lim x → ∞ 4 x 2 − 4 x − 8 x 2 + 4 = 4 1 .
The vertical asymptotes are x = − 1 and x = 2 , and the horizontal asymptote is y = 4 1 . y = 4 1
Explanation
Problem Analysis We are given the function f ( x ) = 4 x 2 − 4 x − 8 x 2 + 4 and asked to find its vertical and horizontal asymptotes.
Finding Vertical Asymptotes Vertical asymptotes occur where the denominator of the function is equal to zero and the numerator is non-zero. Let's find the zeros of the denominator: 4 x 2 − 4 x − 8 = 0 We can simplify this by dividing by 4: x 2 − x − 2 = 0 Factoring the quadratic, we get: ( x − 2 ) ( x + 1 ) = 0 So, the denominator is zero when x = 2 or x = − 1 .
Verifying Vertical Asymptotes Now we need to check if the numerator is non-zero at these points. When x = 2 , the numerator is 2 2 + 4 = 4 + 4 = 8 = 0 . When x = − 1 , the numerator is ( − 1 ) 2 + 4 = 1 + 4 = 5 = 0 . Therefore, the vertical asymptotes are x = 2 and x = − 1 .
Finding Horizontal Asymptote To find the horizontal asymptote, we need to examine the limit of the function as x approaches infinity: x → ∞ lim 4 x 2 − 4 x − 8 x 2 + 4 To evaluate this limit, we divide both the numerator and the denominator by the highest power of x , which is x 2 : x → ∞ lim 4 − x 4 − x 2 8 1 + x 2 4 As x approaches infinity, the terms x 2 4 , x 4 , and x 2 8 all approach zero. Therefore, the limit is: 4 − 0 − 0 1 + 0 = 4 1 So, the horizontal asymptote is y = 4 1 .
Final Answer The vertical asymptotes are x = − 1 and x = 2 , and the horizontal asymptote is y = 4 1 .
Examples
Understanding asymptotes is crucial in fields like physics and engineering. For example, when designing a suspension bridge, engineers need to consider the maximum load the bridge can handle without collapsing. The function describing the bridge's load capacity might have a vertical asymptote, indicating a critical load value that must not be exceeded. Similarly, in drug dosage calculations, the concentration of a drug in the bloodstream over time might approach a horizontal asymptote, representing the maximum effective concentration. Knowing these limits helps in safe and effective drug administration.
