Estimating limits with tables
Before calculating a limit with algebra, it is useful to know how to estimate it with numbers: give $x$ values closer and closer to the point you are interested in and see where the results "point". It is a quick tool for checking a result, or for finding out what is going on when the algebra is not yet obvious.
Tables of values: approaching from both sides
The idea is simple: choose values of $x$ closer and closer to the point, from the left and from the right, and observe $f(x)$.
Observe the table of values of a function $f$ as $x$ approaches $3$ from the left and from the right: | $x$ | $2,9$ | $2,99$ | $2,999$ | $3,001$ | $3,01$ | $3,1$ |…
Observe the table of values of a function $f$ as $x$ approaches $3$ from the left and from the right:
| $x$ | $2,9$ | $2,99$ | $2,999$ | $3,001$ | $3,01$ | $3,1$ |
|---|---|---|---|---|---|---|
| $f(x)$ | $4,80$ | $4,98$ | $4,998$ | $5,002$ | $5,02$ | $5,20$ |
Estimate the value of $\lim_{x \to 3} f(x)$ and explain what you observed in the table to give your answer.
The function $f(x) = \frac{x^2 - 4}{x - 2}$ is not defined at $x = 2$. The table shows its values near that point: | $x$ | $1,9$ | $1,99$ | $1,999$ | $2,001$ | $2,01$ |…
The function $f(x) = \frac{x^2 - 4}{x - 2}$ is not defined at $x = 2$. The table shows its values near that point:
| $x$ | $1,9$ | $1,99$ | $1,999$ | $2,001$ | $2,01$ | $2,1$ |
|---|---|---|---|---|---|---|
| $f(x)$ | $3,9$ | $3,99$ | $3,999$ | $4,001$ | $4,01$ | $4,1$ |
- a) Estimate $\lim_{x \to 2} f(x)$ from the table.
- b) Confirm the result algebraically by simplifying the expression for $f$.
- c) What is the value of $f(2)$? Explain the difference between this fact and the calculated limit.
Limits you cannot "see" by substituting: the gap
Many interesting functions are not defined at exactly the point you are interested in, but that does not stop them from having a limit there.
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: | $x$ | $-0,2$ |…
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:
| $x$ | $-0,2$ | $-0,1$ | $-0,01$ | $0,01$ | $0,1$ | $0,2$ |
|---|---|---|---|---|---|---|
| $f(x)$ | $0,9933$ | $0,9983$ | $1,0000$ | $1,0000$ | $0,9983$ | $0,9933$ |
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.
Be careful with tables: infinities and oscillations
Tables are useful, but they can mislead you if you do not choose the values well or if the function does not behave "calmly" near the point.
Observe the table of values of $f(x) = \frac{1}{(x - 1)^2}$ near $x = 1$: | $x$ | $0,9$ | $0,99$ | $0,999$ | $1,001$ | $1,01$ | $1,1$ | |---|---|---|---|---|---|---| |…
Observe the table of values of $f(x) = \frac{1}{(x - 1)^2}$ near $x = 1$:
| $x$ | $0,9$ | $0,99$ | $0,999$ | $1,001$ | $1,01$ | $1,1$ |
|---|---|---|---|---|---|---|
| $f(x)$ | $100$ | $10.000$ | $1.000.000$ | $1.000.000$ | $10.000$ | $100$ |
- a) Describe the behaviour of $f(x)$ as $x \to 1$.
- b) Does the limit of $f$ exist at $x = 1$? Reason your answer, distinguishing between "infinite limit" and "no finite limit exists".
- c) What does the line $x = 1$ represent for the graph of $f$?
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 $1$".…
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.
The table shows values of $f(x) = \sen\left(\frac{1}{x}\right)$ (in radians) increasingly close to $0$: | $x$ | $0,1$ | $0,01$ | $0,001$ | $0,0001$ |…
The table shows values of $f(x) = \sen\left(\frac{1}{x}\right)$ (in radians) increasingly close to $0$:
| $x$ | $0,1$ | $0,01$ | $0,001$ | $0,0001$ |
|---|---|---|---|---|
| $f(x)$ | $-0,5440$ | $-0,5064$ | $0,8269$ | $-0,3056$ |
- 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.
- 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 limit that defines the number $e$
Some tables point to a number you already know from sequences: the number $e$.
The table shows values of $f(x) = (1 + x)^{1/x}$ near $x = 0$: | $x$ | $0,1$ | $0,01$ | $0,001$ | $0,0001$ | |---|---|---|---|---| | $f(x)$ | $2,5937$ | $2,7048$ |…
The table shows values of $f(x) = (1 + x)^{1/x}$ near $x = 0$:
| $x$ | $0,1$ | $0,01$ | $0,001$ | $0,0001$ |
|---|---|---|---|---|
| $f(x)$ | $2,5937$ | $2,7048$ | $2,7169$ | $2,7181$ |
- a) Estimate the limit to three decimal places.
- b) What well-known number does this limit approach? Write it with its usual letter.
- 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. a) Design the table of values you would use: justify which…
We want to estimate $\lim_{x \to 0} \frac{e^x - 1}{x}$ to three decimal places using only a basic calculator.
- 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.
- b) Construct the table and give the estimation of the limit.
- 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.