Exercises: Calculating limits of functions

The temperature of an electronic component (in degrees Celsius) as a function of time $t$ (in hours) is given by the expression $T(t) = \frac{3t^2 - 5t + 12}{t + 2}$. Calculate the limit of the temperature as $t$ approaches $3$ hours.

Calculate the following algebraic limit by resolving the indeterminate form using factorisation:

$\lim_{x \to 5} \frac{x^2 - 25}{x^2 - 7x + 10}$

Study the left- and right-hand limits of the function $f(x) = \frac{2x + 1}{x - 4}$ as $x$ tends to $4$ and determine whether the overall limit exists at that point.

The velocity of a test vehicle (in m/s) as a function of time $t$ (in seconds) is defined piecewise as $v(t) = 3t + 2$ if $0 \le t < 4$, and $v(t) = t^2 - 2$ if $t \ge 4$. Determine using left- and right-hand limits whether the velocity function is continuous at time $t = 4$.

Resolve the following indeterminate form by factorising the polynomials:

$\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}$

Given the following piecewise function, calculate the left- and right-hand limits at $x = 2$ and determine whether $\lim_{x \to 2} f(x)$ exists:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \ge 2 \end{cases}$

To evaluate the limit $\lim_{x \to 0} (1 + 2x)^{\frac{3}{x}}$, Marcos states the following: “When substituting $x = 0$, the base is $1 + 2 \cdot 0 = 1$. Since the number $1$ raised to any exponent is always $1$, there is no need to make any transformation: the limit is equal to $1$”.

1. Explain with reasoning Marcos's conceptual error, distinguishing between the arithmetic power with a constant base of $1$ and the indeterminate form of type $1^\infty$.

2. Apply the rule for indeterminate forms of type $1^\infty$ recommended in the notes (writing the limit as $e^L$ with $L = \lim_{x \to a} h(x) \cdot (g(x) - 1)$) and find the correct value of the limit.

We want to analyze the behavior of the function $f(x) = \frac{x^2 + x - 6}{x - 2}$ as $x$ approaches $2$.

1. Estimate the result before operating: evaluate the function at points close to $2$ from the left ($x = 1,9$ and $x = 1,99$) and from the right ($x = 2,01$ and $x = 2,1$). From these numerical calculations, conjecture what the value of the limit $\lim_{x \to 2} f(x)$ should be.

2. Check your estimate analytically: verify what indeterminate form is obtained by directly substituting $x = 2$, factor the numerator, and calculate the exact value of the limit.

3. Justify why it is mathematically permissible to cancel the factor $(x - 2)$ when calculating the limit, even though division by zero is not allowed at $x = 2$.

Evaluate the following limit by multiplying the numerator and denominator by the corresponding conjugate expression:

$\lim_{x \to 3} \frac{\sqrt{x + 6} - 3}{x - 3}$

Calculate the value of the limit presenting an indeterminate form of the subtraction of algebraic fractions type:

$\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^2 - 4} \right)$

Determine the value of the following limit by resolving the indeterminate form associated with the number $e$:

$\lim_{x \to +\infty} \left( \frac{x + 5}{x - 2} \right)^{3x}$

An electricity company sets its tariff $T(x)$ in euros based on electricity consumption $x$ in kWh. For consumption less than $100$ kWh, the tariff is $T(x) = 0,15x + 5$. For consumption greater than or equal to $100$ kWh, the tariff changes to $T(x) = \frac{ax^2 + 200}{x}$. Find the value of the parameter $a$ so that the limit of the tariff exists as consumption approaches $100$ kWh.

Calculate the limit at infinity of the following function containing radicals, by analysing the degrees of the numerator and denominator:

$\lim_{x \to +\infty} \frac{\sqrt{9x^2 + 4x} - 2x}{5x + 1}$

Calculate the value of the parameter $k$ so that the following limit equality holds:

$\lim_{x \to +\infty} \frac{kx^3 - 5x + 1}{2x^3 + x^2 - 8} = 4$

Study the existence of the limit at $x = 0$ of the function $f(x) = \frac{|x|}{x} + x^2$ by first calculating its left- and right-hand limits.

The number of registered users on a new streaming platform, expressed in thousands, is modelled by the logistic function $U(t) = \frac{500}{1 + 49 e^{-0,2t}}$, where $t$ is the time in months since its launch. What value does the number of users tend towards over time?

Observe the graph of the function $f(x)$ representing the temperature profile of a chemical process as a function of time $x$. Determine the following limits graphically: $\lim_{x \to 2^-} f(x)$, $\lim_{x \to 2^+} f(x)$ and $\lim_{x \to +\infty} f(x)$.

Consider the limit $\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$, which has an indeterminate form of type $\frac{0}{0}$. To solve it, two students propose different algebraic paths:

• Method A (Lucía): Multiply and divide the fraction by the conjugate of the numerator, $(\sqrt{x} + 2)$.

• Method B (David): Factor the denominator considering it a difference of squares using the identity $x - 4 = (\sqrt{x})^2 - 2^2 = (\sqrt{x} - 2)(\sqrt{x} + 2)$, and then simplify.

• Work through the calculation step by step using both methods and verify that both lead to the same numerical result.

• If you had to calculate instead the limit $\lim_{x \to 4} \frac{\sqrt{2x + 1} - 3}{x - 4}$, justify which of the two previous methods is more advantageous and straightforward, explaining the limitations you would encounter when trying to apply the other method.

On a regulated section of the A-3 highway managed by the Directorate-General for Traffic (DGT), the average vehicular flow speed $v(d)$ (in $\text{km/h}$) is modeled as a function of traffic density $d$ (in vehicles per kilometer) according to the piecewise function:

where $k$ is a real adjustment constant.

1. To avoid sudden braking in traffic when reaching $50\text{ vehicles/km}$, the speed must vary continuously; that is, $\lim_{d \to 50} v(d)$ must exist. Set up the corresponding one-sided limits and equate their expressions to find the value of the parameter $k$.

2. With the obtained value of $k$, calculate $\lim_{d \to +\infty} v(d)$ and interpret the result in the traffic context. Reason whether the value obtained accurately reflects a total traffic jam or gridlock situation, or if it shows that the mathematical model has a limited range of validity.

Calculate the following limit, which presents an indeterminate form of type $\infty - \infty$ involving square roots:

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

Evaluate the following limit with indeterminate form $\frac{0}{0}$ containing two radicals in the numerator:

$\lim_{x \to 0} \frac{\sqrt{4 + x} - \sqrt{4 - x}}{3x}$

Consider a rectangle inscribed in a circle of radius $R = 10$ cm. If $x$ is the length of the base of the rectangle, its area is given by the function $A(x) = x \sqrt{400 - x^2}$. An architect needs to know how the ratio between the area of the rectangle and the square of its base behaves when the base becomes very small. Calculate $\lim_{x \to 0^+} \frac{A(x)}{x^2}$.

The relativistic mass of a moving particle is given by $m(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass, $v$ is the velocity of the particle and $c$ is the speed of light. Calculate the limit of the mass as the velocity $v$ approaches the speed of light $c$ from the left ($v \to c^-$) and interpret the mathematical result in the physical context.

Calculate the values of the parameters $a$ and $b$ so that the following equality holds at infinity:

$\lim_{x \to +\infty} \left( \frac{ax^2 + bx - 1}{x + 2} - 3x \right) = 5$

Solve the following indeterminate form $\infty - \infty$ with radicals of different indices. Pay special attention to the algebraic strategy required to combine or treat the terms separately:

$\lim_{x \to +\infty} \left( \sqrt[3]{x^3 + 6x^2} - \sqrt{x^2 + 2x} \right)$

Given that $\lim_{x \to 0} \frac{\sin x}{x} = 1$, calculate the following trigonometric limit by making the necessary algebraic transformations (hint: multiply the numerator and denominator by the conjugate of the numerator):

$\lim_{x \to 0} \frac{1 - \cos x}{x^2}$

In an optics experiment, a laser beam passes through a medium with a variable refractive index. The deflection of the beam defines a right-angled triangle whose base has length $x$ and whose height is given by the curve $h(x) = \frac{1 - e^{-2x}}{x}$. Look at the diagram and calculate the limit of the area of the shaded triangle $A(x)$ as $x \to 0^+$.

1. Consider the function $f(x) = \left( \frac{x^2 + ax + 1}{x^2 + 2} \right)^{3x}$. Determine with reasoning the value of the non-zero real parameter $a$ such that $\lim_{x \to +\infty} f(x) = e^9$.

2. With the value of the parameter $a$ found, construct a second-degree polynomial $P(x) = c x^2 + d x + e$ that simultaneously satisfies two conditions:

3. Its y-intercept matches this parameter: $P(0) = a$.

4. The limit as $x \to 1$ of the quotient $\frac{P(x)}{x - 1}$ exists and is equal to $4$: $\lim_{x \to 1} \frac{P(x)}{x - 1} = 4$.

Explain why, for the second limit to exist and be finite, it is essential that an indeterminate form $\frac{0}{0}$ is generated (that is, that $P(1) = 0$), and calculate the coefficients $c$ and $d$ from this.

Elena has to evaluate the limit $\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - x \right)$. Her submission contains the following working:

“As $x$ approaches $+\infty$, inside the square root the term $6x$ can be neglected compared to $x^2$. Therefore, $\sqrt{x^2 + 6x} \approx \sqrt{x^2} = x$. Substituting this into the expression we get:

1. Explain the conceptual error Elena makes by truncating or approximating terms in isolation within an indeterminate form of type $\infty - \infty$.

2. Correct the exercise by finding the exact value of the limit using the conjugate technique.

3. Generalize the problem: analyze and discuss with reasoning the value of the limit $\lim_{x \to +\infty} \left( \sqrt{x^2 + 6x} - ax \right)$ depending on the possible values of the real parameter $a$ (distinguishing the cases $a \le 0$, $0 < a < 1$, $a = 1$, and $a > 1$).