Exercises: Limit of a function at a point

Observe the graph of the function $f(x)$ shown below and determine, intuitively, the value of the following one-sided limits at the breakpoint $x = 2$:

1. a) $\lim_{x \to 2^-} f(x)$
2. b) $\lim_{x \to 2^2} f(x)$
3. c) Does the overall limit $\lim_{x \to 2} f(x)$ exist? Justify your answer based on the previous results.

To study the behavior of the function $f(x) = \frac{x^2 - 9}{x - 3}$ near the value $x = 3$ (where the function is undefined), complete the following table of values using your calculator:

$x$	$2,9$	$2,99$	$2,999$	$3,001$	$3,01$	$3,1$
$f(x)$

In light of the results, what appears to be the value of the limit $\lim_{x \to 3} f(x)$?

Calculate the following immediate limits by performing direct substitution into the expression of the function:

1. a) $\lim_{x \to -1} (x^3 - 3x^2 + 5x - 2)$
2. b) $\lim_{x \to 2} \frac{x^2 + 4}{x + 1}$
3. c) $\lim_{x \to 0} \sqrt{2x + 9}$

Given the rational function $g(x) = \frac{x + 2}{x - 4}$, we want to analyse what happens when the variable $x$ approaches $4$.

1. a) Calculate the one-sided limits $\lim_{x \to 4^-} g(x)$ and $\lim_{x \to 4^+} g(x)$.
2. b) What kind of behavior does the graph of the function show in the vicinity of $x = 4$? Use the following diagram to help visualize the trend.

Consider the piecewise function:

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

Calculate the value that the constant $k$ must take for the overall limit of the function to exist at the point $x = 1$. Hint: The one-sided limits must coincide.

A temperature sensor in a chemical reaction records data according to the function $T(t) = \frac{3t + 6}{t + 2}$, where $T$ is the temperature in degrees Fahrenheit and $t$ is the time in minutes.

1. a) Try to calculate the temperature at the exact instant $t = -2$ minutes. What do you observe?
2. b) Calculate the value of the limit $\lim_{t \to -2} T(t)$ by first simplifying the expression of the function. What does this result mean regarding the continuity of the temperature at that point?

Determine the value of the following limits of simple trigonometric and logarithmic functions by direct substitution (make sure the point belongs to the domain):

1. a) $\lim_{x \to \pi} (\cos x + \text{sen } x)$
2. b) $\lim_{x \to e} (\ln x + 2x)$
3. c) $\lim_{x \to 0} \frac{e^x + 1}{2}$

Explain graphically why the function $f(x) = \frac{1}{x^2}$ tends to $+\infty$ as $x$ approaches $0$, from both the left and the right. Draw a sketch of the function highlighting the behavior near the origin.

Calculate the value of the following limit by resolving the indeterminate form through the factorization of the polynomials involved:

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

To solve it, remember that if substituting the value of the variable results in $\frac{0}{0}$, both polynomials share at least one common root. Use synthetic division or algebraic identities to simplify the expression.

Given the following piecewise function, determine the value of the parameter $k \in \mathbb{R}$ so that the limit of the function exists at the critical point $x = 1$:

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

Justify your answer by checking the equality of the one-sided limits.

Calculate the limit of the following irrational function at the point $x = 5$. If you encounter an indeterminate form, use the technique of multiplying by the conjugate expression:

$\lim_{x \to 5} \dfrac{\sqrt{x - 1} - 2}{x^2 - 25}$

Study the behavior of the function $g(x) = \dfrac{2x + 1}{x^2 - 1}$ in the vicinity of its undefined points. To do this, calculate the one-sided limits at $x = 1$ and $x = -1$ and determine whether the function has vertical asymptotes at these values.

A biotechnology laboratory has modeled the concentration of an enzyme in a chemical reaction using the function $C(t)$, where $t$ is the time in minutes:

$C(t) = \dfrac{t^2 - 9}{2t^2 - 5t - 3}$

The scientists are interested in knowing the value of the concentration as the time approaches $3$ minutes, at which point a phase change occurs. Calculate:

$\lim_{t \to 3} C(t)$

Find the value of the coefficients $a$ and $b$ so that the following equality holds, knowing that the limit is finite:

$\lim_{x \to 0} \dfrac{ax^2 + bx}{\sqrt{x + 4} - 2} = 8$

Hint: First analyze what must happen in the numerator for the limit not to be infinite.

Analyze the following graph of the function $h(x)$ and determine, visually, the value of the following limits and of the function at the point:

1. a) $\lim_{x \to -2^-} h(x)$
2. b) $\lim_{x \to -2^+} h(x)$
3. c) $h(-2)$
4. d) $\lim_{x \to 3} h(x)$

Calculate the one-sided limits of the function $f(x) = \dfrac{x^2 - |x - 2| - 4}{x - 2}$ as $x$ tends to $2$. Does the overall limit exist at this point?

Note: Remember to rewrite the function by removing the modulus depending on the interval of approach.

In the design of a ramp for a skatepark, engineers have modelled the curvature of a critical section using the function:

$$f(x) = \frac{\sqrt{x^2 + 5} - 3}{x^2 + ax + b}$$

It is known that for the ramp to be safe and usable, the function must have a finite limit when the value of the horizontal position $x$ approaches $2$. Determine the values of the real parameters $a$ and $b$ knowing that, in addition, the ramp passes through the origin of coordinates $(0,0)$. Once found, calculate the value of this limit.

Calculate the value of the following limit by applying trigonometric identities or the use of equivalent infinitesimals:

$$\lim_{x \to 0} \frac{1 - \cos(3x)}{x \cdot \tan(2x)}$$

Explain with reasoning whether the function has a removable discontinuity at $x = 0$ based on the result obtained.

A communications satellite is adjusting its orbit. The signal strength $S$ at a receiving station depends on an interference factor $x$ according to the expression:

$$S(x) = \left( \frac{x^2 + 3x + 1}{x^2 + 4} \right)^{\frac{1}{x-2}}$$

Due to a processing software failure, the signal is interrupted just as the interference reaches the critical value $x = 2$. Determine what value the signal strength tends to as we approach this critical point.

Given the piecewise function defined by real parameters $k$ and $m$:

$$g(x) = \begin{cases} \frac{e^{kx} - 1}{x} & \text{if } x < 0 \\ m \cdot \cos(x) + 2 & \text{if } x \geq 0 \end{cases}$$

Find the relationship that must exist between $k$ and $m$ for the limit of the function to exist at $x = 0$. What values should these parameters take for this limit to be exactly $5$?

Consider the function $h(x)$ representing the air pressure inside an experimental chamber:

$$h(x) = \frac{|x-1|}{x^2 + x - 2}$$

Analyze the existence of the limit at the point $x = 1$. To do this, calculate the one-sided limits and justify whether the function is convergent at this point. Graphically represent the behavior of the function in the vicinity of $x = 1$.

Find the value of the parameters $a$ and $b$ so that the following equality of limits holds:

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

Use polynomial factorization to resolve the resulting indeterminate form.

Calculate the limit of the function $f(x)$ as $x$ tends to $1$:

$$f(x) = \frac{\sqrt[3]{x} - 1}{\sqrt{x} - 1}$$

Hint: You can use a change of variable of the form $x = t^n$ to simultaneously eliminate both roots.

In an acoustics laboratory, the frequency of a sound wave passing through a special filter is studied. The response of the filter is given by:

$$F(x) = \frac{x^2 - k^2}{\sqrt{x^2 + 7} - \sqrt{k^2 + 7}}$$

where $k$ is a characteristic constant of the material. Calculate the limit of the function $F(x)$ as the variable $x$ tends to $k$. Does the final result depend on the specific value of the constant $k$? Justify your answer.