Limiting Behavior

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limit notation

End-behavior is a simpler approximate description of function values as we move way out in the domain to the very very very large numbers - the tail of the domain. Our phrases for this movement in the domain are

tending to infinity
tending to negative infinity

We also refer to this as limiting behavior.

Our shorthand notation for “the limiting behavior of” is $\lim \limits _{x \to \infty }$. This is placed to the left of the function.

\[ \lim _{x \to -\infty } f(x) \]

We use limit notation to describe end-behavior, when the end-behavior is a constant or unbounded.

$\blacktriangleright $ We have seen limiting or end-behavior of exponential functions.

\[ \lim _{x \to -\infty } 1.5^x = 0 \]

as $x \to \infty $, the function values become unbounded and our notation for that looks like

\[ \lim \limits _{x \to \infty } 1.5^x = \infty \]

$\blacktriangleright $ Same with logaarithms.

\[ \lim \limits _{t \to \infty } \log _2(t+5) = \infty \]

$\infty $ is NOT a real number. The function does reach $\infty $. $\lim \limits _{x \to \infty } f(x) = \infty $ is just notation telling us that all of the values of $f(x)$ become unbounded as we move out along the real line.

That is, for each real number $M$, “eventually” $f(x) > M$.

That is, for each real number $M$, there is a real number $m$ such that $f(x) > M$ for all $x > m$.

$\blacktriangleright $ Some rational functions approach a constant in their end-behavior, which we see on their graphs as a horizontal asymptote.

graph of $y = f(x) = \frac {2x-4}{x+3}$

\[ \lim \limits _{x \to -\infty } f(x) = 2 \]

\[ \lim \limits _{x \to \infty } f(x) = 2 \]

Limits

Let $G(m)$ be defined by the following graph.

\[ \lim _{x \to -\infty } f(x) = \answer {6} \]

\[ \lim _{x \to \infty } f(x) = \answer {-4} \]

Limits

Let $T(y)$ be defined by

\[ T(y) = \frac {2 \sin (3y)}{y} \, \text { defined on } \, (2, \infty ) \]

\[ \lim _{y \to \infty } T(y) = \answer {0} \]

Limits

Let $R(t)$, with its natural domain, be defined by

\[ R(t) = \frac {(2t+5)(t-3)(3t-7)}{(t-8)(t+1)(2t+9)} \, \]

\[ \lim _{\answer {t} \to \infty } R(t) = \answer {3} \]

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more examples can be found by following this link
More Examples of Function Behavior

2025-01-07 01:11:02