Calculating a limit in the EvAU is almost always a matter of applying a fixed procedure: substitute, check whether an indeterminate form appears and, depending on which one it is, apply the technique that resolves it. This section brings together the four indeterminate forms you must master with complete confidence.
The $\frac00$ indeterminate form with polynomials: factorise
If $P(a)=0$ and $Q(a)=0$ in $\dfrac{P(x)}{Q(x)}$, both polynomials share the factor $(x-a)$: by factorising (using Ruffini's rule if necessary) and cancelling, the indeterminate form disappears.
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To understand the concept of a limit, let us analyse the behaviour of the function $f(x) = \frac{x^2 - 4}{x - 2}$ as the values of $x$ get very close to $2$, without…
To understand the concept of a limit, let us analyse the behaviour of the function $f(x) = \frac{x^2 - 4}{x - 2}$ as the values of $x$ get very close to $2$, without being exactly $2$.
Complete the following table of values using your calculator and predict what the value of $\displaystyle \lim_{x \to 2} f(x)$ will be:
Calculate the following limits by analysing the degree of the polynomials and the highest power terms (indeterminate form $\frac{\infty}{\infty}$): a)…
Calculate the following limits by analysing the degree of the polynomials and the highest power terms (indeterminate form $\frac{\infty}{\infty}$):
Radicals: the conjugate resolves $\frac00$ and $\infty-\infty$
When an indeterminate form comes with a root, neither factorising nor comparing degrees works directly: you have to multiply and divide by the conjugate, which turns the root into a square.
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In the analysis of the spread of news on a social network, the function $P(t) = \dfrac{100}{1 + 20 \cdot e^{-0,5t}}$ is used, where $P$ is the percentage of the…
In the analysis of the spread of news on a social network, the function $P(t) = \dfrac{100}{1 + 20 \cdot e^{-0,5t}}$ is used, where $P$ is the percentage of the population aware of the news and $t$ is the time in hours.
a) Calculate $\lim_{t \to \infty} P(t)$ and interpret the result in the context of the problem.
b) If the rate of spread is given by the derivative (which we will study later, but suppose you are given the simplified expression for relative growth), analyse what happens to the difference $100 - P(t)$ as $t \to \infty$.
c) Determine, with reasoning, whether the function has any vertical asymptotes.
The $\frac\infty\infty$ indeterminate form: comparing degrees
When the numerator and denominator both tend to infinity, the result depends only on comparing their degrees (if they are polynomials) or, in general, on identifying which term dominates.
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Determine the value of the parameter $k \in \mathbb{R}$ so that the limit of the function $f(x)$ exists at the point $x = 1$, given the function: $$f(x) = \begin{cases}…
Determine the value of the parameter $k \in \mathbb{R}$ so that the limit of the function $f(x)$ exists at the point $x = 1$, given the function:
When the base of a power tends to $1$ and the exponent tends to infinity, you cannot substitute directly: the result is expressed using the number $e$.
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Given the rational function $f(x) = \dfrac{x^3 + ax^2 + bx + 2}{x^2 - 1}$, determine the values of the real parameters $a$ and $b$ knowing that the finite limit of the…
Given the rational function $f(x) = \dfrac{x^3 + ax^2 + bx + 2}{x^2 - 1}$, determine the values of the real parameters $a$ and $b$ knowing that the finite limit of the function exists as $x$ tends to $1$ and that this limit is exactly $L = -2$. Once you have found the parameters, justify with reasoning whether the function has a vertical asymptote at $x = -1$.