Exercises: Infinite branches in 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?

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.

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?

A language learning mobile app is rapidly gaining users. The number of active users, in thousands, is modelled by the exponential function $U(t) = 5 \cdot 1,2^t$, where $t$ is the time in months since launch.

a) Calculate the limit of $U(t)$ as $t \to +\infty$ and explain what type of infinite branch the function presents (does it have a horizontal asymptote at that end?).

b) Study what happens when $t \to -\infty$. Does the function have any horizontal asymptote in that direction? State its equation.

c) Sketch the graph of the function in your notebook, showing the infinite branches at both ends.

You can do it on paper and send a photo with your answer.

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 $x \to -\infty$.

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.

Calculate the following limits of elementary functions:

a) $\displaystyle \lim_{x \to +\infty} \log_{10}(x^2 + 1)$

b) $\displaystyle \lim_{x \to -\infty} e^{x+2}$

c) $\displaystyle \lim_{x \to 0^+} \frac{1}{\ln(x)}$

Observe the graph of the function $f(x)$, which combines rational and exponential behavior. The dashed lines are reference lines and each tick mark on the axes equals 1 unit. From the graph, identify the values of the indicated limits and write the equations of its horizontal and vertical asymptotes.

Calculate, reasoning from the observation:

1. $\lim_{x \to +\infty} f(x)$

2. $\lim_{x \to -\infty} f(x)$

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

4. The equations of the horizontal and vertical asymptotes of $f(x)$, justifying them with the previous limits.

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?

Study the asymptotes and the behaviour at the infinite branches of the logarithmic function:

$$f(x) = \ln(x^2 - 16)$$

To do this:

1. Find the domain of definition of the function.

2. Calculate the one-sided limits at the endpoints of the intervals of the domain.

3. Determine if there is any horizontal asymptote as $x \to +\infty$.

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.

Consider the piecewise function:

$$h(x) = \begin{cases} 2^x & \text{if } x \le 1 \\ \log_{10}(x) + k & \text{if } x > 1 \end{cases}$$

1. Calculate the value of $k$ so that the function is continuous at $x = 1$.

2. Once $k$ is found, study the limits of the function at infinity: $\lim_{x \to -\infty} h(x)$ and $\lim_{x \to +\infty} h(x)$.

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.

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 $\lim_{t \to +\infty} T(t)$.

Determine all the asymptotes of the function:

$$f(x) = \frac{x \cdot \ln(x)}{x-1}$$

To do this, calculate the necessary limits as $x \to 1^+$ (endpoint of the domain due to the denominator becoming zero) and as $x \to +\infty$. Does this function have any parabolic branch or horizontal asymptote?

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:

$$U(t) = \frac{120}{2 + 8e^{-0,15t}}$$

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.

Study all the infinite branches (vertical, horizontal and oblique asymptotes) of the composite logarithmic function defined by:

$$f(x) = \ln\left(\frac{x^2 - 9}{x + 1}\right)$$

Note that you must first determine the domain of the function, as the argument of the logarithm must be strictly positive. Describe the behaviour of the function at the boundaries of its domain using one-sided limits.

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.

Calculate the value of the following limit, which leads to an indeterminate form of type $1^\infty$ related to the number $e$:

$$\lim_{x \to +\infty} \left(\frac{2\ln(x) + 5}{2\ln(x) - 1}\right)^{\ln(x)}$$

Determine the values of the parameters $a$ and $b$ such that the following function is continuous on its entire domain and has a horizontal asymptote at $y = 4$ as $x \to -\infty$:

$$f(x) = \begin{cases} \frac{ax + 10}{x - 2} & \text{if } x < 0 \\ b \cdot e^x + 1 & \text{if } x \geq 0 \end{cases}$$

Explain the steps taken in detail, first analyzing the limit at infinity and then the continuity at the transition point.

Given the rational function with exponential terms:

$$h(x) = \frac{3^x - 3^{-x}}{3^x + 3^{-x}}$$

Calculate its horizontal asymptotes by finding the limits $\lim_{x \to +\infty} h(x)$ and $\lim_{x \to -\infty} h(x)$. Be very careful with the behaviour of $3^x$ and $3^{-x}$ depending on whether $x$ tends to plus or minus infinity.

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.

Analyse the existence of limits and vertical asymptotes in the function $f(x) = \frac{1}{\ln(x^2 - 3)}$.

In particular, study what happens at the values of $x$ that make the denominator equal to zero. Determine the one-sided limits at those points and describe whether the function approaches $+\infty$ or $-\infty$. Are there intervals where the function is not defined? Justify your answer.