Exercises: Behaviour at infinity

Look at the following graph of the function $f(x)$. Determine the value the function approaches when $x$ becomes infinitely large ($x \to +\infty$) and when it becomes infinitely small ($x \to -\infty$).

The following table shows the values taken by a function $g(x)$ for increasingly larger values of $x$:

• $x = 10 \rightarrow g(x) = 2,9$

• $x = 100 \rightarrow g(x) = 2,99$

• $x = 1000 \rightarrow g(x) = 2,999$

• $x = 10000 \rightarrow g(x) = 2,9999$

In light of these results, what value does the function $g(x)$ tend to when $x \to +\infty$?

Calculate the behavior at infinity of the following polynomial function, that is, find the value of the limit:

$\lim_{x \to +\infty} (2x^3 - 5x^2 + 7x - 1)$

Determine the trend of the function $h(x) = -4x^5 + 3x^2 - 8$ when $x$ tends to $-\infty$. To do this, calculate $\lim_{x \to -\infty} h(x)$.

Calculate the following limit of a rational function where the degree of the denominator is greater than that of the numerator:

$\lim_{x \to +\infty} \frac{3x - 5}{x^2 + 4x + 1}$

A colony of bacteria grows according to the function $P(t) = 150 \cdot 2^t$, where $t$ is measured in hours. What happens to the bacteria population when time advances indefinitely ($t \to +\infty$)?

The amount of a radioactive substance in milligrams follows the function $M(t) = 500 \cdot 0,8^t$, where $t$ is the years elapsed. What value does the amount of substance tend to in the very long term ($t \to +\infty$)?

Look at the graph of the function $C(x)$ representing the average cost (in dollars) of manufacturing $x$ objects. State the equation of the horizontal asymptote and explain its meaning when the number of objects produced tends to infinity.

The average production cost of $x$ units of an electronic item is given by the rational function $C(x) = \frac{2500 + 15x}{x}$ in dollars. Calculate the limit of this function when $x \to +\infty$ and interpret the result in the context of the company.

The temperature in degrees Fahrenheit of a freshly poured cup of coffee in a room decreases with time $t$ (in minutes) according to the exponential function $T(t) = 22 + 68 \cdot 0,85^t$. What temperature does the coffee tend to when a long time has elapsed? Justify your answer by mathematically calculating the corresponding limit.

In an isolated ecosystem, the population of a certain bird species is modeled by the function $P(t) = \frac{1200t + 400}{2t + 5}$, where $t$ is the time in years since records began. Determine what value the bird population will stabilize at in the long term.

Calculate the limit when $x \to -\infty$ of the following rational function, paying attention to the signs and degrees of the polynomials:

$\lim_{x \to -\infty} \frac{5x^3 - 2x^2 + x}{4 - 3x^3}$

Study the behavior of the polynomial function $f(x) = (2x^2 - 3x + 1) - (x^3 - 5x)$ by calculating its limits when $x \to +\infty$ and when $x \to -\infty$.

Find the value of the parameter $k$ knowing that the rational function $f(x) = \frac{kx^2 - 3x + 7}{4x^2 + 5}$ has a horizontal asymptote at the line $y = 3$.

Given the following piecewise function:

$f(x) = \begin{cases} \frac{4x^2 + 1}{x^2 - 3} & \text{if } x > 0 \\ 3^x + 2 & \text{if } x \leq 0 \end{cases}$

Calculate the limits $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.

A principal of $\$2000$ is invested at an annual compound interest rate of $4\%$. The function describing the accumulated capital after $t$ years is $C(t) = 2000 \cdot 1,04^t$. What is the limit of $C(t)$ as $t \to +\infty$? Do you consider this mathematical model to be realistic for extremely long periods of time? Explain your answer.

The concentration of a certain drug in the blood (in mg/L) $t$ hours after its oral administration follows the function $C(t) = \frac{30t}{t^2 + 9}$. Calculate what value the concentration approaches in the long run. Next, set up and solve an inequality to determine after what exact hour the blood concentration will be strictly less than $1$ mg/L.

Determine the values of the parameters $a$ and $b$ of the rational function $f(x) = \frac{ax + 8}{bx - 2}$ knowing that its graph passes through the point $(1, -10)$ and has a horizontal asymptote at the line $y = 3$.

Study the behavior at infinity of the function containing an absolute value $f(x) = \frac{4x + |x|}{x + 2}$. To do this, calculate the limits as $x \to +\infty$ and as $x \to -\infty$. Does the function have the same horizontal asymptote on the right as on the left?

A software company estimates that its annual profit (in thousands of dollars) as a function of advertising expenses $x$ (also in thousands of dollars) is modeled by the function $B(x) = \frac{120x^2 - 500}{3x^2 + 2x + 10}$. Is there a maximum limit for profits no matter how much is invested in advertising indefinitely? Justify your answer by calculating the limit at infinity and state how many dollars this cap amounts to.

Let the rational function be $f(x) = \frac{3x}{x+2}$. Calculate its limit $L$ as $x \to +\infty$. Next, set up and solve an inequality to find for what values of $x > 0$ it is true that the distance between $f(x)$ and its limit $L$ is strictly less than $0,01$.

Consider the functions $f(x) = x^4$ and $g(x) = 1,5^x$. For small values of $x$, the polynomial function is greater than the exponential one (for example, for $x=2$, $16 > 2,25$). However, determine which of the two grows faster at infinity by using a table of values for $x = 10, 20$ and $30$. From your conclusion, deduce the value of the limit $\lim_{x \to +\infty} \frac{x^4}{1,5^x}$.

Look at the following graph representing a function $f(x)$ with two different horizontal asymptotes. From the graph, deduce the value of $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$. Based on what you have studied about exponential functions, write a possible algebraic expression that exhibits this type of double asymptotic behavior.

A saline solution begins to be poured into an industrial tank containing $500$ gallons of pure water at a rate of $10$ gallons per minute, with a concentration of $40$ ounces of salt per gallon. At the same time, the well-mixed solution leaves the tank at the same rate to keep the volume constant. The salt concentration in the tank (in oz/gal) at time $t$ (in minutes) is given by the function $C(t) = 40(1 - e^{-0,02t})$. Calculate the limit of $C(t)$ as $t \to +\infty$ and explain the physical meaning of this result.