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Section 11.3 Limits at Infinity
To take the limit of a function as
\(x\) becomes infinitely large, Maple recognizes
infinity and
-infinity. These can be used to find horizontal asymptotes. If the function does not have a horizontal asymptote, the limit may result in
\(\pm \infty\text{.}\)
> g(x) := (3*x^2 + 5*x - 10) / (5*x^2 - 20*x + 1);
\begin{equation*}
\displaystyle g\, := \,x\mapsto {\frac {3\,{x}^{2}+5\,x-10}{5\,{x}^{2}-20\,x+1}}
\end{equation*}
> limit(g(x), x=infinity);
\begin{equation*}
\displaystyle 3/5
\end{equation*}
An oscillating function such as
\(\sin(x)\) may not have a definable limit. Maple will attempt to determine a range for the minimum and maximum of the function.
\begin{equation*}
\displaystyle h\, := \,x\mapsto \sin \left( x \right)
\end{equation*}
Aside Since
\(\sin(x)\) oscillates between
\(-1\) and
\(1\text{,}\) Maple cannot determine a unique value for the limit as
\(x \rightarrow -\infty\text{.}\)
> limit(h(x), x=-infinity);
\begin{equation*}
\displaystyle {-1\ldots 1}
\end{equation*}
Example 11.2 . Horizontal Asymptotes and Limits at Infinity.
In this example, we will examine the function
\begin{equation*}
f(t) = \frac{2000}{1+{ e}^{-t+2}}\text{,}
\end{equation*}
which is known as a logistic function. Logistic functions have many applications, such as population modeling.
> logistic(t) := 2000/(1 + exp(-t+2));
\begin{equation*}
\displaystyle logistic\, := \,t\mapsto \frac{2000}{1+{{ e}^{-t+2}}}
\end{equation*}
Judging by the plot of the logistic function, it appears that the function may have horizontal asymptotes. To find the right-hand asymptote, we take the limit as
\(t \rightarrow \infty\text{.}\) Here we are using
t=infinity rather than
x=infinity, since the variable of this function is
\(t\text{.}\)
> limit(logistic(t), t=infinity);
\begin{equation*}
\displaystyle 2000
\end{equation*}
To find the left-hand asymptote, we take the limit as
\(t \rightarrow -\infty\text{.}\)
> limit(logistic(t), t=-infinity);
\begin{equation*}
\displaystyle 0
\end{equation*}
