Find and interpret the horizontal asymptote

Question

a persons temperature T, in degrees Farenheit,during an illness is given by the function T(t) = 98.6t^2+ 4t + 98.6/ t^2+1 where time t is given in hours since then onset of the illness. Find and interpret the horizontal asymptote.

Answer 1

Given:

T(t) = (98.6t² + 4t + 98.6) / (t² + 1) where t is time in hours.

Firstly, lets solve it algebraically. To find the horizontal asymptote, we can apply the "Leading Term Test" which is where we compare the highest degree (highest power) in the numerator and denominator. In this case, the degree of the numerator is 2 and the degree of the demominator is also 2. Since the degree is same in both numerator and denominator, in the case we take the ratio of the leading coefficients. The leading coefficient is the number in front of the highest degree variable. So, the leading coefficient in numerator is 98.6 since that number is in front of t² and the leading coefficient in denominator is 1 since that is in front of t² so we divide them:

98.6 / 1 = 98.6

Therefore, y = 98.6 is the horizontal asymptote. This means that as time passes, the person's body temperature approaches 98.6°F, which is normal body temperature.

This can also be solved by applying calculus. We can take the limit of the function as t approaches infinity:

lim T(t) = lim (98.6t² + 4t + 98.6) / (t² + 1)

t→∞ t→∞

To solve this, we can divide every term in the numerator and denominator by t²:

= lim (98.6t²/t² + 4t/t² + 98.6/t²) / (t²/t² + 1/t²)

t→∞

We let t→∞ so:

4/t → 0,

98.6/t² → 0,

1/t² → 0

So, we end up with:

lim T(t) = 98.6 / 1 = 98.6

t→∞

Therefore, y = 98.6 is the horizontal asymptote. This means that as time passes, the person's body temperature approaches 98.6°F, which is normal body temperature.

Answer 2

T(t) = (98.6t² + 4t + 98.6) / (t² + 1)

Notice that the order of both the numerator and denominator is 2.

So, the horizontal asymptote will be the ratio of the coeffecient of t².

Thus, the horizontal asymptote is

T = 98.6

When you graph this equation, you will see that it never crosses T = 98.6.

This means the person's temperature will never be below 98.6 degrees Farenheit.

*Please feel free to ask me any question for clarification.