Sure, let's delve into what a horizontal asymptote is and how it applies to this particular function.
A horizontal asymptote refers to a horizontal line that the graph of a function approaches as the input (typically [tex]\( t \)[/tex] in this context) grows indefinitely large. For the function [tex]\( T(t) = \frac{4t}{t^2 + 1} \)[/tex], we want to find the behavior of [tex]\( T(t) \)[/tex] as [tex]\( t \)[/tex] approaches infinity.
Here are the steps to determine the horizontal asymptote:
1. Degrees of Polynomials:
- The given function is [tex]\( T(t) = \frac{4t}{t^2 + 1} \)[/tex].
- The degree of the polynomial in the numerator (4t) is 1.
- The degree of the polynomial in the denominator ([tex]\( t^2 + 1 \)[/tex]) is 2.
2. Comparing Degrees:
- When the degree of the numerator is less than the degree of the denominator in a rational function, the horizontal asymptote is [tex]\( y = 0 \)[/tex]. This is because the value of the fraction [tex]\( \frac{4t}{t^2 + 1} \)[/tex] gets smaller and smaller as [tex]\( t \)[/tex] becomes very large (it approaches zero).
3. Interpreting the Horizontal Asymptote:
- The function [tex]\( T(t) \)[/tex] represents the increase in a person's body temperature above [tex]\( 98.6^\circ F \)[/tex].
- If [tex]\( T(t) \)[/tex] approaches 0 as [tex]\( t \)[/tex] becomes very large, then the increase in body temperature above the baseline of [tex]\( 98.6^\circ F \)[/tex] is approaching 0.
- Therefore, the person's body temperature is approaching [tex]\( 98.6^\circ F \)[/tex].
So, the meaning of the horizontal asymptote for the function [tex]\( T(t) = \frac{4t}{t^2 + 1} \)[/tex] is:
The horizontal asymptote of [tex]\( y = 0 \)[/tex] means that the person's temperature will approach [tex]\( 98.6^\circ F \)[/tex] as time elapses.
Hence, the correct interpretation is:
- The horizontal asymptote of [tex]\( y = 0 \)[/tex] means that the person's temperature will approach [tex]\( 98.6^\circ F \)[/tex] as time elapses.
