Estimating Limits with 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 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.

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.

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.

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

1. a) Describe the behavior 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 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.

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

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.

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.

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.

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.