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:

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

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 pesos based on electricity consumption $x$ in kWh. For consumption less than $100$ kWh, the tariff is $T(x) = 3x + 100$. For consumption greater than or equal to $100$ kWh, the tariff changes to $T(x) = \frac{ax^2 + 4\,000}{x}$. 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 $\lim_{x \to +\infty} f(x)$.

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 $x \to 0^+$.