Exercises: Estimating limits using tables

Observe the table of values of a function $f$ as $x$ approaches $3$ from the left and from the right:

Estimate the value of $\lim_{x \to 3} f(x)$ and explain what you observed in the table to give your answer.

The table shows the values of a function $g$ near $x = 1$:

Estimate the left- and right-hand limits of $g$ at $x = 1$ and reason whether $\lim_{x \to 1} g(x)$ exists.

The table contains values of $h(x) = \frac{3x + 1}{x}$ for increasingly large values of $x$:

Estimate $\lim_{x \to +\infty} h(x)$ and confirm the result by calculating the limit algebraically.

The table shows values of $f(x) = \frac{\sen x}{x}$ (with $x$ in radians) around $x = 0$, where the function is not defined, as suggested by the figure:

Estimate $\lim_{x \to 0} \frac{\sen x}{x}$ and explain why it makes sense to talk about a limit even though $f(0)$ does not exist.

Marcos wants to estimate the limit $\lim_{x \to 0} \frac{|x|}{x}$ using a table of values. To do this, he evaluates the function at $x = 0{,}1$, $x = 0{,}01$, and $x = 0{,}001$, obtaining the value $1$ in all three cases. Marcos concludes in his notebook: "Since the values clearly stabilize at $1$ as we approach $0$, the limit exists and is equal to $1$".

1. a) Point out the conceptual error that Marcos makes in his estimation procedure using tables.
2. b) Construct the corresponding table to approach $0$ from the left by evaluating at $x = -0{,}1$, $x = -0{,}01$, and $x = -0{,}001$.
3. c) Based on the results of both trends, reason what can be stated about the existence and value of the limit $\lim_{x \to 0} \frac{|x|}{x}$.

We want to study the behavior of the function $g(x) = \frac{\sqrt{x} - 2}{x - 4}$ near $x = 4$. For this purpose, the following table of values is available:

1. a) Based on the trends observed from the left and from the right in the table, give a reasoned estimate of the value of $\lim_{x \to 4} g(x)$ and express it as an irreducible fraction.
2. b) Check the accuracy of your estimate by simplifying the function: multiply the numerator and the denominator of $g(x)$ by the conjugate $(\sqrt{x} + 2)$ and calculate the limit algebraically.
3. c) If you tried to evaluate $g(4)$ directly by substituting into the original formula, what would you obtain? Explain why this fact does not contradict the result obtained for the limit.

The function $f(x) = \frac{x^2 - 4}{x - 2}$ is not defined at $x = 2$. The table shows its values near that point:

1. a) Estimate $\lim_{x \to 2} f(x)$ from the table.

1. b) Confirm the result algebraically by simplifying the expression for $f$.

1. c) What is the value of $f(2)$? Explain the difference between this fact and the calculated limit.

Observe the table of values of $f(x) = \frac{1}{(x - 1)^2}$ near $x = 1$:

1. a) Describe the behaviour of $f(x)$ as $x \to 1$.

1. b) Does the limit of $f$ exist at $x = 1$? Reason your answer, distinguishing between "infinite limit" and "no finite limit exists".

1. c) What does the line $x = 1$ represent for the graph of $f$?

The position of a moving object is given by $s(t) = t^2 + 2t$ metres.

1. a) Calculate the average velocity in the interval $[2, 2 + h]$ and complete the table:

1. b) Estimate the instantaneous velocity at time $t = 2$ by observing the table.

1. c) Check the result by calculating $s'(2)$ using the definition or differentiation rules.

To estimate $\lim_{x \to 0} \frac{|x|}{x}$, a student only calculates the values $f(0,1) = 1$, $f(0,01) = 1$ and $f(0,001) = 1$, and concludes: "the limit is $1$". Explain the error in their method, calculate the values they should have checked and state the correct conclusion regarding the existence of that limit.

A test vehicle from the DGT conducts a controlled braking test on a straight stretch of the A-6 motorway. The distance traveled (in meters) since the brake is pressed is modeled by $s(t) = 30t - 2{,}5t^2$, where $t$ is the elapsed time in seconds.

To estimate the instantaneous velocity of the vehicle at exactly $2$ seconds, the telemetry sensors calculate the average velocity over increasingly smaller intervals around $t = 2$, using the quotient:

The table shows the measurements from the system:

1. a) Physically interpret what it means to approach $t$ to $2$ from the left ($t < 2$) and from the right ($t > 2$) in this test.
2. b) From the table, give a reasoned estimate of the value of $\lim_{t \to 2} v_m(t)$ and convert this instantaneous velocity to $\text{km/h}$.
3. c) If the on-board computer tried to calculate the velocity by evaluating directly at $t = 2$ using the expression $\frac{s(2) - s(2)}{2 - 2}$, what mathematical difficulty would it encounter? Justify why the concept of limit allows engineers to solve this problem.

Consider the family of rational functions $f(x) = \frac{x^2 + kx - 6}{x - 2}$, where $k$ is a real constant to be determined.

1. a) Reasonably deduce what value $k$ must have so that $\lim_{x \to 2} f(x)$ is a finite real number instead of resulting in a vertical asymptote.
2. b) With the found value of $k$, make a table of values by calculating $f(x)$ for $x \in \{1{,}9;\; 1{,}99;\; 2{,}01;\; 2{,}1\}$ and numerically estimate the value of the limit.
3. c) If we now modify the numerator and define the function $h(x) = \frac{x - 5}{x - 2}$, analyze the signs of the numerator and denominator as $x$ approaches $2$ from both sides, and reason how it would be reflected in a table that $\lim_{x \to 2^-} h(x) = +\infty$ and $\lim_{x \to 2^+} h(x) = -\infty$.

The table shows values of $f(x) = \sen\left(\frac{1}{x}\right)$ (in radians) increasingly close to $0$:

1. a) A student claims: "since the values of $x$ approach $0$, the limit exists and is approximately $-0,13$ (the mean of the table)". Critique their reasoning.

1. b) Calculate $f\left(\frac{1}{k\pi}\right)$ for $k = 1, 2, 3, \ldots$ and reason, choosing suitable sequences of values for $x$, that the limit does not exist.

We want to estimate $\lim_{x \to 0} \frac{e^x - 1}{x}$ to three decimal places using only a basic calculator.

1. a) Design the table of values you would use: justify which values of $x$ you choose and why it is advisable to approach from both sides.

1. b) Construct the table and give the estimation of the limit.

1. c) Check that your estimation is consistent with the fact that this limit is, by definition, the derivative of $e^x$ at $x = 0$.

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

The table shows values of $f(x) = (1 + x)^{1/x}$ near $x = 0$:

1. a) Estimate the limit to three decimal places.

1. b) What well-known number does this limit approach? Write it with its usual letter.

1. c) Explain why it is not enough to calculate $f(0)$ to answer.

When a processor's fan turns off, its temperature $T(t)$, in degrees Celsius, begins to rise. The average rates of change of temperature have been measured over increasingly small intervals of $[60, 60 + h]$ seconds:

1. a) Estimate the instantaneous rate of change of temperature at time $t = 60\ \text{s}$.

1. b) What concept from the topic of derivatives does this value represent? Write it using mathematical notation.

1. c) Interpret the sign of the rate in context and explain what information a negative rate would provide.

We want to determine the value of $\lim_{x \to 0} (1 + x)^{2/x}$. In class, two different approaches are proposed to solve it:

• Method 1 (Numerical estimation with a table): Evaluate the function at symmetric values increasingly close to $0$ ($x = \pm 0{,}1$; $\pm 0{,}01$; $\pm 0{,}001$) and observe what number the decimals stabilize at.

• Method 2 (Algebraic properties): Transform the expression using powers as $\left[(1+x)^{1/x}\right]^2$ and apply the known reference limit $\lim_{x \to 0} (1+x)^{1/x} = e \approx 2{,}71828$.

1. a) Apply Method 1: calculate the values for $x = 0{,}01$ and $x = -0{,}01$ using your calculator (rounding to four decimal places) and propose an estimate for the limit.
2. b) Apply Method 2: find the exact value of the limit in terms of $e$, calculate its decimal approximation to five significant figures, and compare this result with the estimate obtained in the previous section.
3. c) Critically assess both methods: which of the two provides mathematical certainty and which only a conjecture? What problem would Method 1 present if the analyzed function exhibited strongly oscillatory behavior near $x = 0$?

David wants to estimate the limit $\lim_{x \to 0} \frac{\sqrt{x^2 + 100} - 10}{x^2}$. Convinced that "the closer $x$ is to $0$, the more reliable the estimate will be", he enters extremely small values into his scientific calculator and records the results in this table:

David concludes: "At first it seemed to approach $0{,}05$, but when really pushing it with $10^{-6}$ and $10^{-8}$ the function gives $0$. Therefore, the limit is $0$".

1. a) Uncover the flaw in David's reasoning. Explain why a scientific calculator returns exactly $0$ for $x = 10^{-8}$ even though the numerator $\sqrt{(10^{-8})^2 + 100} - 10$ is not mathematically zero (think about significant figures and the machine's rounding).
2. b) Multiply the numerator and denominator by the conjugate expression $\sqrt{x^2 + 100} + 10$, simplify algebraically, and calculate the exact value of the limit.
3. c) Based on what you have learned, reason why the class notes recommend reducing the distance to $a$ step by step, order of magnitude by order of magnitude ($0{,}1$; $0{,}01$; $0{,}001$), instead of hastily jumping to values like $10^{-8}$ when tabulating with digital tools.