Infinite limits and limits at infinity
There are two ways for a function to "go off to infinity": it can do so as it approaches a specific point (the curve shoots up or down vertically) or when $x$ grows or decreases without bound (the curve stretches out horizontally). Telling these two situations apart —and calculating what happens in each— is the aim of this section.
Infinite limit at a point: vertical asymptote
When the denominator of a rational function becomes zero at $x_0$ without the numerator doing so, the function "shoots off" near $x_0$.
Given the function $f(x) = \frac{4}{x - 3}$, study its behaviour as $x$ approaches $3$ from both sides. To do so, calculate: a) $\lim_{x \to 3^-} f(x)$ b)…
Given the function $f(x) = \frac{4}{x - 3}$, study its behaviour as $x$ approaches $3$ from both sides. To do so, calculate:
- a) $\lim_{x \to 3^-} f(x)$
- b) $\lim_{x \to 3^+} f(x)$
State whether the function has any vertical asymptote and write its equation.
Complete the following table of values for the function $f(x) = \frac{x^2 - 1}{x - 1}$ and intuitively estimate the value of the limit $\lim_{x \to 1} f(x)$. Explain…
Complete the following table of values for the function $f(x) = \frac{x^2 - 1}{x - 1}$ and intuitively estimate the value of the limit $\lim_{x \to 1} f(x)$. Explain why, even though the function is not defined at $x = 1$, we can still talk about its limit at that point.
| $x$ | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 |
|---|---|---|---|---|---|---|
| $f(x)$ |
Limits at infinity of rational functions
When $x\to\pm\infty$, the only thing that matters in a fraction of polynomials is comparing the degree of the numerator with that of the denominator.
A market study analyzes the evolution of the number of users of a new messaging app, measured in thousands, by means of the function $U(t) = \dfrac{40t + 10}{2t + 5}$,…
A market study analyzes the evolution of the number of users of a new messaging app, measured in thousands, by means of the function $U(t) = \dfrac{40t + 10}{2t + 5}$, where $t$ represents the time in months since its launch. The figure shows the graph of this function.
- a) Analyze what happens to the number of users around the third month. To do so, calculate the following limit, showing your reasoning:
- b) Determine the long-term behavior of the app. Is there a maximum value that the number of users approaches? Interpret the result in terms of horizontal asymptotes.
Calculate the following limits at infinity using the comparison of dominant terms and explain their graphical meaning (whether they tend to a specific value or display…
Calculate the following limits at infinity using the comparison of dominant terms and explain their graphical meaning (whether they tend to a specific value or display parabolic branches):
- a) $\lim_{x \to +\infty} \dfrac{3x^3 - 5x + 2}{1 - 2x^3}$
- b) $\lim_{x \to -\infty} \dfrac{x^2 + 10}{x - 5}$
- c) $\lim_{x \to +\infty} \dfrac{\sqrt{x^4 + 3x}}{5x^2 - 1}$
Indeterminate form $\infty-\infty$ with roots: the conjugate
When a subtraction of terms tends to $\infty-\infty$ and a root appears, substituting does not work: you have to multiply and divide by the conjugate to turn the root into a square.
A forest surveillance drone follows a path whose height in meters above sea level is given by the expression $H(t) = \sqrt{t^2 + 8t} - t$, where $t$ is the time in…
A forest surveillance drone follows a path whose height in meters above sea level is given by the expression $H(t) = \sqrt{t^2 + 8t} - t$, where $t$ is the time in minutes since it starts its patrol. The figure shows its path during the first 20 minutes.
Calculate the limit of the height as time tends to infinity:
$$\lim_{t \to +\infty} (\sqrt{t^2 + 8t} - t)$$At what height does the drone level off its flight in the long run? Resolve the indeterminate form by multiplying by the conjugate.
Growth hierarchy: exponential, power and logarithm
When an exponential, a power of $x$ and a logarithm compete in the same limit, no algebraic calculation is needed: a fixed hierarchy decides which one "wins".
Calculate the value of the following limit involving a difference of irrational functions at infinity:…
Calculate the value of the following limit involving a difference of irrational functions at infinity:
$\lim_{x \to +\infty} \left( \sqrt{4x^2 + 5x - 1} - \sqrt{4x^2 - 3x + 2} \right)$
Explain with reasons why we cannot simply say that the result is $0$ when subtracting two infinities of the same order.