Exercises: Evaluating limits

Given the graph of the function $f(x)$ shown below, visually identify the value of the following limits:

1. a) $\lim_{x \to 2^-} f(x)$

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

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

Calculate the limit at infinity of the following polynomial function, which represents the evolution of the profits of a tech startup (in thousands of pesos) over time $x$:

$$\lim_{x \to +\infty} ( -120x^3 + 3 000x^2 - 6 000 )$$

State which is the dominant term and justify why the sign of the leading coefficient determines the end behavior of the function.

A video game server manages the number of connected players using the function $P(x) = \frac{6x^2 + 20}{2x^2 + 5}$, where $x$ is the time in hours since the launch of an event. Calculate the limit of players as time tends to infinity:

$$\lim_{x \to +\infty} \frac{6x^2 + 20}{2x^2 + 5}$$

In a chemistry laboratory, the concentration of a reactant in a mixture (in mg/l) follows the function $C(t) = \frac{5t + 8}{t^2 + 2}$, where $t$ is the time in minutes. Determine the long-term value to which the concentration tends by calculating:

$$\lim_{t \to +\infty} \frac{5t + 8}{t^2 + 2}$$

Calculate the one-sided limits of the function $f(x) = \frac{3}{x - 5}$ at the point where the denominator vanishes. To do so, study the behavior of $f(x)$ as you approach from the left and from the right, paying attention to the sign of the numerator and the denominator:

1. a) $\lim_{x \to 5^-} \frac{3}{x - 5}$

1. b) $\lim_{x \to 5^+} \frac{3}{x - 5}$

The diagram shows the axes, the vertical asymptote $x=5$ and the direction in which we approach it from each side:

Resolve the indeterminate form $\frac{0}{0}$ by simplifying the fraction through factorising the polynomials (you can use algebraic identities or Ruffini's rule):

$$\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 2x}$$

Based on comparing orders of infinity, determine the value of the following limits without performing complex operations:

1. a) $\lim_{x \to +\infty} \frac{e^x}{x^{10}}$

1. b) $\lim_{x \to +\infty} \frac{\ln(x)}{x^2}$

1. c) $\lim_{x \to +\infty} \frac{2^x}{3^x}$

The altitude of a reconnaissance drone (in meters) with respect to the horizontal distance $x$ covered is given by the function $h(x) = \frac{x^2 - 9}{x - 3}$. Although the drone cannot physically be at position $x = 3$ due to a software restriction, calculate the limit of the altitude as it approaches that point:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$

A precision sensor in a nanotechnology laboratory measures the attractive force between two particles. The signal intensity $S$ as a function of the distance $x$ (in nanometers) is given by the following function:

$$S(x) = \frac{\sqrt{x+5} - 3}{x-4}$$

The figure shows the graph of $S(x)$: for $x=4$ the function is not defined (point with the empty circle).

Calculate the value the signal approaches as the distance $x$ gets closer to $4$ nanometers. Explain which mathematical technique you used to resolve the indeterminate form.

In the study of news propagation on social networks, it is estimated that the number of users reached $U$ (in thousands) after $t$ hours follows the expression:

$$U(t) = \frac{5t^2 - 3t + 2}{2t^2 + \sqrt{t^4 + 1}}$$

Calculate the limit of users reached as time increases indefinitely ($t \to +\infty$). Does the function have a horizontal asymptote? Justify your answer by comparing the degrees of the dominant terms.

Calculate the following limit by resolving the indeterminate form $\infty - \infty$ using the conjugate expression:

$$\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - x \right)$$

Sketch a graph showing what this result means regarding the distance between the radical function and the line $y = x$.

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

Given the rational function depending on a parameter $k \in \mathbb{R}$:

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

Find the value of $k$ so that the limit $\lim_{x \to 2} f(x)$ is finite. Once this value is found, calculate the value of the limit.

The growth of a bird population in a protected nature reserve is governed by the following idealised continuous compound growth law:

$$P(t) = \left( \frac{3t + 5}{3t - 2} \right)^{2t}$$

Calculate the limit of the population as time $t$ tends to infinity. Express the result in terms of the number $e$.

Calculate the value of the following limits by comparing the orders of magnitude at infinity of the functions involved (exponentials, powers, and logarithms):

1. a) $\lim_{x \to +\infty} \frac{\ln(x^{10} + 5)}{x}$

1. b) $\lim_{x \to +\infty} \frac{3^x}{x^2 + 100x}$

1. c) $\lim_{x \to +\infty} \frac{e^x + x^5}{2e^x - \sqrt{x}}$

Consider the piecewise function $f(x)$:

$$f(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{if } x < 3 \\ \sqrt{2x + a} & \text{if } x \ge 3 \end{cases}$$

Calculate the value of the parameter $a$ so that the limit of the function exists at $x = 3$. What is the value of this limit?

Calculate the limit at $x = 0$ of the following trigonometric expression, using equivalent infinitesimals or known trigonometric identities:

$$\lim_{x \to 0} \frac{x \times \text{tg}(3x)}{\cos(x) - 1}$$

A multiplayer video game server has detected that the number of active users, $N(t)$, in thousands, follows an exponential growth model as a function of time $t$ (in months) defined by the expression:

$$N(t) = \left( \frac{3t^2 + 5t + 1}{3t^2 + 2t} \right)^{2t}$$

Calculate the limit of active users as time tends to infinity ($t \to +\infty$). Does the player population stabilise at a finite value or grow indefinitely?

When designing a new ramp for a skatepark, engineers need to ensure that the transition between two sections of the track is smooth (continuous). The height of the ramp $h(x)$ in meters is given by the following function:

$$h(x) = \begin{cases} \dfrac{\ln(1 + kx)}{x} & \text{if } x > 0 \\ \cos(x) + a & \text{if } x \le 0 \end{cases}$$

Determine the values of the parameters $k$ and $a$ such that $\displaystyle \lim_{x \to 0} h(x)$ exists and is equal to the initial height of the ramp, which is $2$ meters.

Calculate the value of the following limit using the comparison of infinities or L'Hôpital's Rule, analysing in detail the order of magnitude of each term:

$$\lim_{x \to +\infty} \frac{e^x + x^{10}}{2e^x - \ln(x^5 + 1)}$$

Which term dominates the growth of the function in the numerator and which in the denominator?

A satellite describes a trajectory whose radial distance to the center of a planet, in thousands of kilometers, is given by $d(x) = \sqrt{x^2 + ax} - \sqrt{x^2 - 1}$, where $x$ represents the mission time. Given that after a long time ($x \to +\infty$) the distance stabilizes at $4$ thousand kilometers, determine the value of the orbital constant $a$.

Give reasons to determine the value of the parameters $a$ and $b$ (where $a, b \in \mathbb{R}$) such that the following equality of limits holds:

$$\lim_{x \to 0} \frac{ax^2 + bx + 1 - e^x}{x^2} = 3$$

Study the behavior of the function $f(x) = \left( \cos x \right)^{1/x^2}$ as $x$ approaches zero. To do so, calculate:

$$\lim_{x \to 0} (\cos x)^{1/x^2}$$

Explain why this limit leads to an indeterminate form of the type $1^\infty$ and solve it using natural logarithms or the specific formula for this type of limit.

Let the function be $g(x) = \dfrac{x^3 - 2x^2 + k}{x^2 - 4}$.

1. a) Calculate the value of $k$ so that the limit of the function as $x \to 2$ is finite.

1. b) Once you have found $k$, calculate that limit.

1. c) Classify the type of discontinuity that the function has at $x = 2$ and at $x = -2$.

Calculate the limit of the following sequence, which frequently appears in continuous compound interest and probability problems:

$$\lim_{n \to \infty} \left( \sqrt{n^2 + n} - n \right)^n$$

Note: This exercise requires careful treatment of the inner indeterminacy before evaluating the overall power.