The concept of a limit
In 1st Bachillerato you worked with the limit as an intuitive idea: "where a function is heading". In the university entrance exam (EvAU) you are expected to do more: to handle the definition precisely, to justify why a limit exists or not, and never to confuse it with the value of the function at that point. First of all, it is worth reviewing something that this whole unit will use constantly: the domain.
The domain: where it makes sense to ask about the limit
To talk about the limit of $f$ at $x=a$ it is not essential for $a$ to belong to the domain of $f$, but you do need to be able to identify the forbidden values: those that make a denominator zero, those that make the argument of an even-index root negative, or those that make the argument of a logarithm non-positive.
Analyse the domain of definition of the following complex function, which combines logarithmic and rational constraints under a square root:…
Analyse the domain of definition of the following complex function, which combines logarithmic and rational constraints under a square root:
$f(x) = \sqrt{\frac{\ln(x^2 - 8)}{x - 5}}$
Express the domain using interval notation. Reason how each component (the logarithm, the denominator and the root) affects the exclusion of values on the real line.
The intuitive idea of a limit
As in 1st Bachillerato, intuition needs no formulae: it is enough to observe what happens to $f(x)$ when $x$ gets very close to a value $a$, without having to reach it.
A marketing analyst is studying the growth in followers of an Instagram account. The function $f(x)=3x$ represents the number of followers (in thousands) $x$ hours after…
A marketing analyst is studying the growth in followers of an Instagram account. The function $f(x)=3x$ represents the number of followers (in thousands) $x$ hours after a campaign is launched. To understand what happens right at the fourth hour, complete the following table of values by approaching $x = 4$ and predict the value of the limit $\lim_{x \to 4} f(x)$.
| $x$ | $3,9$ | $3,99$ | $3,999$ | $4,001$ | $4,01$ | $4,1$ |
|---|---|---|---|---|---|---|
| $f(x)$ |
- a) Calculate the missing values of $f(x)$ in the table (you can give up to three decimal places).
- b) Observe which value $f(x)$ gets closer to as $x$ approaches $4$ from the left and from the right, and predict the value of $\lim_{x \to 4} f(x)$.
Formal definition: neighbourhoods of $a$ and of $L$
The idea of "getting close" is made precise using the language of neighbourhoods: a neighbourhood of a point is simply an open interval centred on it. The EvAU does not require you to handle $\varepsilon$ and $\delta$ with algebraic fluency, but you do need to understand what they mean.
Look at the following graph, which represents the charge level of a smart battery $C(t)$ (in %) as a function of time $t$ (in minutes). Due to a software error, the…
Look at the following graph, which represents the charge level of a smart battery $C(t)$ (in %) as a function of time $t$ (in minutes). Due to a software error, the sensor does not record the exact value at minute $t = 3$ (hollow point), although the charge seems to follow a clear trend. Use the graph to determine the following values:
- a) $\lim_{t \to 3^-} C(t)$ (limit as we approach 3 from the left).
- b) $\lim_{t \to 3^+} C(t)$ (limit as we approach 3 from the right).
- c) Does the two-sided limit $\lim_{t \to 3} C(t)$ exist? Justify your answer.
Given the function $f(x)$ shown in the following graph, analyse the existence of the limit at the critical points. To do this: 1. State the value of the one-sided limits…
Given the function $f(x)$ shown in the following graph, analyse the existence of the limit at the critical points. To do this:
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State the value of the one-sided limits when $x \to -1$ and when $x \to 2$.
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Determine whether the overall limit exists at those points.
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Explain the difference between the value of the function $f(2)$ and the value of $\lim_{x \to 2} f(x)$.
The limit and the value of the function: two different questions
$\displaystyle\lim_{x\to a} f(x)$ and $f(a)$ answer different questions: one looks at the neighbourhood of $a$, the other at the exact point. Confusing them is one of the most frequent mistakes in the EvAU.
A market study indicates that the profit of a new mobile app (in thousands of euros) follows the function $B(t) = \frac{2t^2 - 8}{t - 2}$, where $t$ is the time in…
A market study indicates that the profit of a new mobile app (in thousands of euros) follows the function $B(t) = \frac{2t^2 - 8}{t - 2}$, where $t$ is the time in months since its launch.
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What happens to the profit as the time approaches 2 months? Make a numerical approximation by calculating $B(1,9)$, $B(1,99)$, $B(2,01)$ and $B(2,1)$.
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Although the function is not defined at $t = 2$, could you assign a logical value to the profit at that instant based on the trend? Justify your answer using the concept of limit.
Given the function $f(x) = \frac{|x-1|}{x-1}$: 1. Calculate the one-sided limits at the point $x = 1$. 2. Based on the previous results, reason whether the overall limit…
Given the function $f(x) = \frac{|x-1|}{x-1}$:
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Calculate the one-sided limits at the point $x = 1$.
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Based on the previous results, reason whether the overall limit exists at $x = 1$.
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Is it possible to redefine the function at $x = 1$ so that it is continuous? Justify your answer based on the type of discontinuity.