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 break point $x = 2$:

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

To study the behaviour 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 direct limits by substituting directly into the expression for 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 $\lim_{x \to 4^+} g(x)$.

1. b) What kind of behaviour does the graph of the function show near $x = 4$? Use the following sketch of the graph to visualise the trend, and justify your answer with the results from part a).

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 be equal.

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 Celsius 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 lies within 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 $0$, both from the left and from the right. Draw a sketch of the function highlighting the behavior near the origin.

You can do it on paper and send a photo with your answer.

Alejandro has to evaluate the following limit on a 2nd year of Bachillerato exam:

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

On his exam paper he writes the following:

"I substitute $x = 0$ and get $\frac{e^0 - 0 - 1}{0 + 0} = \frac{0}{0}$. Since it is an indeterminate form, I apply L'Hôpital's rule by differentiating the numerator and the denominator:

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

Since this gives $\frac{0}{0}$ again, I apply L'Hôpital a second time:

$$\lim_{x \to 0} \frac{e^x}{2} = \frac{e^0}{2} = \frac{1}{2}$$"

1. a) Pinpoint Alejandro's mistake precisely and explain which essential condition of L'Hôpital's rule he has violated.
2. b) Correct the procedure and find the actual value of the limit.
3. c) Based on the correct result, indicate whether the graph of the function $f(x) = \frac{e^x - x - 1}{x^2 + 2x}$ has a vertical asymptote or a removable discontinuity at $x = 0$, justifying your answer.

We wish to study the behavior of the function $f(x) = \frac{e^{2x} - 1}{\sin(3x)}$ near the origin.

1. a) Estimate: before applying any formal rule, evaluate mentally or using a calculator the value of $f(x)$ for the nearby points $x = 0,05$ and $x = -0,05$ (remember that the trigonometric argument is measured in radians). From these two numerical values, conjecture what the exact value of $\lim_{x \to 0} f(x)$ should be.
2. b) Verify: calculate $\lim_{x \to 0} f(x)$ analytically by applying l'Hôpital's rule, explicitly checking beforehand that the conditions for its applicability are met.
3. c) Compare the numerical estimate with the exact result obtained and reflect: why does a table of numerical values alone not constitute a sufficient mathematical proof of the existence of the limit?

Calculate the value of the following limit by resolving the indeterminate form through the factorisation 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 yields $\frac{0}{0}$, both polynomials share at least one common root. Use Ruffini's rule or algebraic identities to simplify the expression.

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

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

Justify your answer by verifying that the one-sided limits are equal.

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 behaviour of the function $g(x) = \dfrac{2x + 1}{x^2 - 1}$ near the points where it is undefined. To do this, calculate the one-sided limits at $x = 1$ and $x = -1$ and determine whether the function has vertical asymptotes at those values.

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

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

Scientists are interested in finding the concentration value as time approaches $3$ minutes, the point at which 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 and that also $a + b = 5$:

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

Analyse the following graph of the function $h(x)$ and visually determine the value of the following limits and the value 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 that point?

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

The task is to calculate the value of:

$$\lim_{x \to 0} \frac{\sqrt{1 + 4x} - (1 + 2x)}{x^2}$$

1. a) Solve it using algebraic methods by multiplying and dividing by the conjugate expression of the numerator until the indeterminacy is removed.
2. b) Solve it by applying l'Hôpital's rule, clearly indicating at which steps you verify that its hypotheses are satisfied and how many times you need to differentiate.
3. c) Compare both approaches: which of the two methods is safer and less prone to calculation errors in this specific case? Furthermore, justify which of the two methods you would choose if the numerator contained the cube root $\sqrt[3]{1 + 4x}$ instead of the square root.

During a winter thermal inversion episode in Madrid, air quality technicians analyze the concentration of nitrogen dioxide ($\text{NO}_2$) recorded at a monitoring station after the municipal protocol of traffic restrictions is activated. The average concentration $C(t)$, expressed in micrograms per cubic meter ($\mu\text{g/m}^3$), is modeled as a function of elapsed time $t$ (in hours, with $t > 0$) using the continuous function:

$$C(t) = \frac{400 \cdot (1 - \cos(0,5t)) + 180t}{t^2 + 2t}$$

1. a) Calculate $\lim_{t \to 0^+} C(t)$ by applying l'Hôpital's rule and properly justifying its use, to determine the theoretical value of the concentration right at the beginning of the intervention.
2. b) European Directive 2008/50/EC establishes an hourly limit value for health protection of $200\ \mu\text{g/m}^3$ and an alert threshold for the population of $400\ \mu\text{g/m}^3$. In light of the calculated limit, were any of these thresholds exceeded at the instant $t \to 0^+$?
3. c) Calculate $\lim_{t \to +\infty} C(t)$ and interpret what this mathematical model predicts in the long term regarding the presence of the pollutant if the restrictions were maintained indefinitely.

When designing a ramp for a skate park, engineers have modeled the curvature of a critical section using the function:

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

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

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

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

Give a reasoned explanation of whether the function has a removable discontinuity at $x = 0$ based on the result obtained.

A communications satellite is adjusting its orbit. The signal intensity $S$ 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 fault in the processing software, the signal drops out just as the interference reaches the critical value $x = 2$. Determine the value that the signal intensity tends to as we approach this critical point.

Given the piecewise function defined in terms of 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 relation that must exist between $k$ and $m$ for the limit of the function at $x = 0$ to exist. What values should these parameters take for this limit to be exactly $5$?

Consider the function $h(x)$ that represents 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 so, calculate the one-sided limits and justify whether the function is convergent at that point. Graph the behavior of the function near $x = 1$, indicating in your drawing the values that $h(x)$ approaches when $x \to 1^-$ and when $x \to 1^+$.

You can do it on paper and send a photo with your answer.

Find the values of the parameters $a$ and $b$ such that the following equality of limits holds:

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

Use polynomial factorisation to resolve the indeterminate form generated.

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

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

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 constant characteristic 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.

Consider the function dependent on two real parameters $a, b \in \mathbb{R}$:

$$f(x) = \frac{a\cos x + bx + e^x - 1}{x^2}$$

It is known that the finite limit $\lim_{x \to 0} f(x) = L$ exists, with $L \in \mathbb{R}$.

1. a) Prove with reasoning, using the algebra of limits and the continuity of the functions involved, that for the limit to be a finite number it is a mandatory condition that $\lim_{x \to 0} (a\cos x + bx + e^x - 1) = 0$. What would happen geometrically to the line $x = 0$ if this limit of the numerator were different from zero?
2. b) By applying the previous condition and l'Hôpital's rule as many times as strictly necessary, determine the unique values of the parameters $a$ and $b$, and calculate the exact value of the finite limit $L$.
3. c) If the extended function $g(x)$ is defined such that $g(x) = f(x)$ for $x \neq 0$ and $g(0) = L$, what type of discontinuity did $f$ have at $x = 0$? Justify your answer according to the usual classification of discontinuities.

Lucía tries to calculate the following limit in class:

$$\lim_{x \to 0} \frac{x^2 \sin\left(\frac{1}{x}\right)}{\sin x}$$

In her exam, she develops the following argument:

"In the numerator, as $x \to 0$, we have the product of $x^2 \to 0$ and a bounded function $\sin(1/x)$, which tends to $0$. In the denominator, $\sin 0 = 0$. It is an indeterminate form $\frac{0}{0}$, so I apply l'Hôpital's rule by differentiating the numerator and the denominator:

$$\lim_{x \to 0} \frac{2x\sin\left(\frac{1}{x}\right) + x^2\cos\left(\frac{1}{x}\right)\cdot\left(-\frac{1}{x^2}\right)}{\cos x} = \lim_{x \to 0} \frac{2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right)}{\cos x}$$

When calculating this last limit, the term $2x\sin(1/x) \to 0$, but $\cos(1/x)$ oscillates without limit between $-1$ and $1$ as $x \to 0$, while the denominator tends to $\cos 0 = 1$. Since the limit of the quotient of derivatives does not exist, I conclude that the original limit does not exist either."

1. a) Explain the logical and theoretical error that Lucía makes in stating her conclusion. Clearly distinguish whether l'Hôpital's rule expresses a necessary condition or a sufficient condition.
2. b) Calculate the actual value of the original limit using a valid method that does not use l'Hôpital's rule (for example, by rewriting the expression as a convenient product and applying properties of bounded functions or the fundamental trigonometric limit).
3. c) In light of the correct value, classify the discontinuity of the function at $x = 0$ and explain how it can be redefined to make it continuous at that point.