Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.68

Chapter 2, Problem 2.5.68

If a function f represents a system that varies in time, the existence of lim ${$\displaystyle\[\lim$_{t$\rightarrow\]\infty$}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

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Identify the function that describes the system: p(t) = $\frac{1500}{3 + 2e^{-0.1t}$}.

To determine if a steady state exists, evaluate the limit of p(t) as t approaches infinity: $\lim$_{t $\to$ $\infty$} p(t).

Observe that as t approaches infinity, the term e^{-0.1t} approaches zero because the exponential function decays to zero.

Substitute e^{-0.1t} with 0 in the expression for p(t) to simplify the limit: $\lim$_{t $\to$ $\infty$} $\frac{1500}{3 + 2 \cdot 0}$.

Calculate the simplified expression to find the steady-state value: $\frac{1500}{3}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value, which can be finite or infinite. In this context, the limit as t approaches infinity indicates how the function behaves as time progresses indefinitely. Understanding limits is crucial for analyzing the long-term behavior of dynamic systems, such as population models.

Steady State (Equilibrium)

A steady state, or equilibrium, occurs when a system's variables no longer change over time, meaning that the system has reached a stable condition. In mathematical terms, this is often represented by the limit of a function as time approaches infinity being a constant value. Identifying steady states is essential in various fields, including biology and economics, to predict long-term outcomes.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value, often modeled by functions involving the exponential function e. In the given population model, the term e^{-0.1t} represents the decay of the influence of initial conditions over time, which is critical for determining how the population stabilizes as time progresses.

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Related Practice

Textbook Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

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Textbook Question

Determine the following limits.

$\lim_{x \to 0^{-}} \csc (x)$

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Textbook Question

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$ , determine ${$\displaystyle\]\lim$_{x$\to$ a^{+}}}f$\left$(x$\right$)$ , ${$\displaystyle\]\lim$_{x$\to$ a^{-}}}f$\left$(x$\right$)$ , and ${$\displaystyle\]\lim$_{x$\to$ a}}f$\left$(x$\right$)$ .

$f (x) = \frac{x + 1}{x^{3} - 4 x^{2} + 4 x}$

Textbook Question

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

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Textbook Question

Evaluate each limit.

lim x→2 √4x+10 / 2x−2

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Textbook Question

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.