Exercises: Calculation of limits

Calculate the value of the following limit by direct substitution:

$\lim_{x \to 2} \frac{3x^2 - x + 1}{x + 3}$

Determine the limit at infinity of the following polynomial, paying attention to the signs:

$\lim_{x \to -\infty} (-4x^3 + 2x^2 - 5x + 1)$

Calculate the limit of the following rational function as $x$ tends to infinity:

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

Find the value of the following limit by observing the degrees of the numerator and the denominator:

$\lim_{x \to +\infty} \frac{3x - 7}{x^3 + 2x^2 - 5}$

Resolve the following indeterminate form by factorizing the polynomials:

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

Calculate the one-sided limits of the function at $x = 1$ and state whether the ordinary limit exists:

$\lim_{x \to 1} \frac{2x}{x - 1}$

Determine the value of the following exponential limits based on the value of their bases:

$\lim_{x \to +\infty} \left(\frac{4}{3}\right)^x$ and $\lim_{x \to -\infty} \left(\frac{2}{5}\right)^x$

Given the following piecewise function, calculate the one-sided limits at $x = 2$ and determine if $\lim_{x \to 2} f(x)$ exists:

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

Resolve the following indeterminate form of type $\infty - \infty$ by first operating on the algebraic fractions:

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

Calculate the following limit by multiplying the numerator and denominator by the appropriate conjugate expression:

$\lim_{x \to 4} \frac{\sqrt{x} - 2}{x^2 - 16}$

Determine the value of the following limit, which presents an indeterminate form of type $1^\infty$, using the formula for the number $e$:

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

The concentration of a certain drug in a patient's blood, measured in milligrams per liter, varies with time $t$ (in hours) according to the function $C(t) = \frac{15t}{t^2 + 4}$. Calculate the long-term concentration, that is, evaluate the limit as $t \to +\infty$ and interpret the result.

The number of registered users on a new streaming platform, expressed in thousands, is modeled 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 to over time?

Find the value of the parameter $k$ so that the following limit exists and is a real number, and then calculate this limit:

$\lim_{x \to 2} \frac{x^2 + kx - 6}{x - 2}$

Resolve the following indeterminate form with radicals at infinity by multiplying by the conjugate:

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

Observe the graph of the function $f(x)$ which represents 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 values of the parameters $a$ and $b$ so that the following equality holds at infinity:

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

According to the theory of special relativity, the relativistic mass of an object moving at a speed $v$ is given by $m(v) = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $m_0$ is the rest mass and $c$ is the speed of light in a vacuum. Calculate the limit of the relativistic mass as the speed of the object approaches the speed of light ($\lim_{v \to c^-} m(v)$) and interpret mathematically why an object with mass cannot reach this speed.

Solve the following limit by applying the properties of logarithms to simplify the expression before evaluating the tendency at infinity:

$\lim_{x \to +\infty} \frac{\ln(x^3 + 2x) - \ln(x^2 - 1)}{\ln(5x) - \ln(x + 3)}$

Determine the value of the parameter $a$ so that the following exponential limit results in $e^4$:

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

The total cost of producing $x$ units of a high-precision electronic component is given by the function $\$C(x) = 5000 + 12x + 0.05x^2$. The average cost per unit is $C_m(x) = \frac{C(x)}{x}$. Calculate the limit of the average cost as the number of units produced tends to infinity and explain what this result means for the viability of the company on a large scale.

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

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

Knowing that $\lim_{x \to 0} \frac{\sin x}{x} = 1$, calculate the following trigonometric limit by performing 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 deviation of the beam defines a right triangle whose base measures $x$ and whose height is given by the curve $h(x) = \frac{1 - e^{-2x}}{x}$. Observe the diagram and calculate the limit of the area of the shaded triangle $A(x)$ as $x \to 0^+$.