2. Horizontal Asymptotes

Using an example model of Newton’s law of cooling, we figure out the equation:

$T(t)=T_{env}+(T(0)-T_{env})e^{-rt}$

So now if our refrigerator is $5°$ Celsius and our object is $20°$ Celsius when placed in, we get the equation:

$T(t)=5+(20-5)e^{-t}$

There is a horizontal asymptote of $y=5$ where it only approaches and it doesn’t touch.

Horizontal Asymptote

Definition

The function $f(x)$ has a horizontal asymptote at $y=a$ if

$|f(x)-a|\to 0$ as $x\to \infty$ or $x\to -\infty$

Note:

A function can have $0$, $1$ or $2$ distinct horizontal asymptotes.

Limits

Notation

If $y=a$ is a horizontal asymptote for $f(x)$ when $x\to \infty$ we write

${\lim }_{x\to \infty }f(x)=a$

If $y=a$ is a horizontal asymptote for $f(x)$ when $x\to -\infty$ we write

${\lim }_{x\to -\infty }f(x)=a$