Infinite branches of exponential and logarithmic functions

A mold colony grows in a Petri dish in a school laboratory. The area it covers, in $cm^2$, follows the model of an exponential function that depends on time $t$ in days: $M(t) = 2 \cdot 1,5^t$. The graph shows the function for $t$ between 0 and 6 days.

a) What value does the area covered by the mold approach if the colony keeps growing indefinitely? Express this fact using limit notation.

b) Look at the graph of the function and describe the type of infinite branch it presents when $t \to +\infty$.

The Wi-Fi signal strength in a coffee shop, measured in microwatts ($\mu$W), decreases as we move away from the antenna according to the function $I(d) = 60 \cdot e^{-0,1d}$, where $d$ is the distance in metres.

a) Calculate the limit of the signal strength as the distance $d$ tends to $+\infty$.

b) What does the result obtained mean in the context of the problem? Is there a horizontal asymptote?

Find the horizontal asymptotes, if any, of the following functions by studying their limits at infinity:

a) $f(x) = 5^{x} + 2$

b) $g(x) = \left(\frac{1}{2}\right)^{x} - 3$

Note: Remember to study the limit both as $x \to +\infty$ and as $x \to -\infty$.

Given the logarithmic function $f(x) = \ln(x - 4)$, find:

a) The domain of the function.

b) Study the behavior of the function as $x$ approaches $4$ from the right ($x \to 4^+$) and determine whether there is a vertical asymptote.

Given the function $f(x) = e^{1/(x-3)}$, calculate the following one-sided limits:

1. $\lim_{x \to 3^-} f(x)$

2. $\lim_{x \to 3^+} f(x)$

In light of the results, indicate whether there is a vertical asymptote at $x = 3$ and describe the behaviour of the function on each side of the point.

A logistics company estimates that the maintenance cost of its fleet of lorries, in euros, follows the model $C(x) = 1500 + \log(x + 10)$, where $x$ is the number of kilometres driven.

Analyse the infinite branch of the function as the mileage increases indefinitely ($x \to +\infty$). Does the maintenance cost have a maximum limit or does it grow without bound?

Match each of the following limit expressions with the corresponding graphical feature of a logarithmic function $f(x) = \log_a(x)$ with $a > 1$:

1. $\displaystyle \lim_{x \to 0^+} f(x) = -\infty$

2. $\displaystyle \lim_{x \to +\infty} f(x) = +\infty$

A) The function has a slow-growing parabolic branch.

B) The function has a vertical asymptote at the y-axis.

Analyse the behaviour of the function $g(x) = 2 + \frac{\ln(x)}{x}$ as the independent variable grows indefinitely ($x \to +\infty$).

Explain whether the function has any horizontal asymptote and represent it schematically on a coordinate axis.

A market study estimates that the number of subscribers to a new streaming platform, in thousands of people, follows the function $S(t) = \frac{120}{1 + 4 \cdot e^{-0,1t}}$, where $t$ is the time in months since its launch.

Determine the limit of subscribers that the platform will have "in the long term", that is, when $t \to +\infty$. What is this behaviour called from the point of view of the infinite branches of the function?

In a chemistry lab, the temperature of a reaction (in $^\circ C$) is given by the function $T(t) = 20 + 80 \cdot e^{-0,5t}$, where $t$ is the time in minutes. The figure shows the graph of the function during the first 10 minutes.

1. a) What is the initial temperature of the reaction?

1. b) At what temperature will the mixture stabilize if we let a long time pass? Calculate $\lim_{t \to +\infty} T(t)$.

The impact of a digital marketing campaign for a new mobile app is measured by the number of daily active users, $U(t)$, in thousands, where $t$ is the time in days since launch. The model is given by the following function:

The figure shows the graph of $U(t)$ over the first 40 days.

Analyze the long-term behavior of the user community by calculating $\lim_{t \to +\infty} U(t)$ and interpret the meaning of the horizontal asymptote you find. What would happen to the limit if the exponent were positive, that is, if $e^{0,15t}$ appeared instead of $e^{-0,15t}$? Justify your answer.

In a class debate about rates of growth, a student claims that, since the numerator of the function $g(x) = \frac{x}{e^x}$ tends to infinity as $x \to +\infty$, the function $g(x)$ cannot have a finite limit at infinity either.

Prove by calculating $\lim_{x \to +\infty} \frac{x}{e^x}$ whether the student is correct or not, comparing the rate of growth of the exponential function with that of the linear function. Does $g(x)$ have a horizontal asymptote? State its equation.

An educational content channel earns a monthly advertising income (in hundreds of euros) given by the function $R(x) = 50 \cdot \ln(x+1) - 0,5x$, where $x$ represents the months elapsed since its launch.

1. a) Calculate $\lim_{x \to +\infty} R(x)$ by comparing the orders of magnitude of logarithmic growth versus linear growth.
2. b) Is this a good model to represent the channel's long-term income? Justify your answer based on the infinite branch obtained.