Exercises: Limit of a function at a point

Calculate the following limit by directly substituting the value of $x$:

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

Observe the graph of the function $f(x)$ and determine the one-sided limits $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$. From these, state whether $\lim_{x \to 2} f(x)$ exists.

Given the piecewise function:

$f(x) = \begin{cases} 3x - 2 & \text{if } x < 1 \\ x^2 & \text{if } x > 1 \end{cases}$

Calculate the one-sided limits at $x = 1$ and determine whether the limit exists at that point.

Solve the following limit which presents an indeterminate form of the type $0/0$:

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

Calculate the one-sided limits of the following function at $x = 1$ to determine the sign of infinity:

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

Calculate the limit of the following function with a modulus at a point where its internal argument is not zero:

$\lim_{x \to -2} |x^2 + 4x + 3|$

Solve the following limit by multiplying the numerator and denominator by the conjugate expression:

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

Observe the following table of values for a function $f(x)$ in the neighbourhood of $x = 3$. Deduce the value of $\lim_{x \to 3} f(x)$.

Solve the following indeterminate form by factoring the polynomials (you can use Ruffini's rule):

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

Calculate the following limit by solving the indeterminate form using the conjugate expression:

$\lim_{x \to 5} \frac{\sqrt{x - 1} - 2}{x - 5}$

Determine the value of the constant $k$ so that the limit of the function $f(x)$ at $x = 2$ exists and coincides with the value of $f(2)$:

$f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\ k & \text{if } x = 2 \end{cases}$

The temperature $T$ (in $^\circ\text{C}$) of an electronic component as a function of voltage $v$ (in volts) is given by the expression $T(v) = \frac{3v^2 - 12}{v - 2}$ for $v \neq 2$. What value does the temperature tend to when the voltage approaches $2\text{ V}$?

Study the existence of the limit of the following function at the point where the modulus is zero, previously calculating its one-sided limits:

$\lim_{x \to 4} \frac{|x - 4|}{x - 4}$

Calculate the following one-sided limit of a logarithmic function. Justify the sign of the resulting infinity:

$\lim_{x \to 3^+} \ln(x - 3)$

The area of a triangle depends on a parameter $x$ according to the function $A(x) = \frac{x^2 - 25}{x - 5}$. Calculate the limit of the area when the parameter $x \to 5$.

Analyse the attached graph of the function $f(x)$. Determine the value of the limits $\lim_{x \to -1} f(x)$ and $\lim_{x \to 2} f(x)$, stating whether they are finite, infinite or do not exist.

Solve the following limit which presents an indeterminate form of the type $0/0$ with roots in the numerator and in the denominator:

$\lim_{x \to 2} \frac{\sqrt{x + 2} - 2}{\sqrt{2x + 5} - 3}$

Find the values of the parameters $a$ and $b$ so that the limits of the function $f(x)$ exist at the points of change of definition $x = 0$ and $x = 2$:

$f(x) = \begin{cases} ax + b & \text{if } x < 0 \\ x^2 - 1 & \text{if } 0 \le x < 2 \\ bx - a & \text{if } x \ge 2 \end{cases}$

Determine the value of the parameter $k$ so that the following limit is a finite real number, and calculate the value of this limit:

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

The marginal profit of a company, in thousands of dollars, when producing $x$ tons of a product is given by the function $B(x) = \frac{1000x - 1000}{x^2 - 1}$. What value does the marginal profit tend to when production approaches $1\text{ tonelada}$?

Calculate the following trigonometric limit using the fundamental identities to resolve the indeterminate form:

$\lim_{x \to \pi/2} \frac{1 - \sin(x)}{\cos^2(x)}$

The position of a particle as a function of time $t$ is given by $s(t) = \frac{\sqrt{t + 9} - 3}{t}$. Calculate the instantaneous velocity at the instant $t = 0$, which is defined as the limit of the position divided by the time as $t \to 0$.

Solve the following limit by operating on the algebraic fractions to resolve the indeterminate form $\infty - \infty$:

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

A point $P(x, f(x))$ moves along the parabola $f(x) = x^2$. A secant line is drawn passing through the origin of coordinates $O(0,0)$ and the point $P$. Calculate the limit of the slope of this secant line as the $x$-coordinate of the point $P$ tends to $0$.