Tutorial: Limits and continuity

Go to Part A: Calculating limits algebraically

This tutorial: Part B: Calculating limits at infinity

(This topic is also in Section 10.3 in Finite Mathematics and Applied Calculus)

Some limits at infinity

In %%partAtut we looked at limits of the form $\displaystyle \lim_{x \to a} f(x)$ where $a$ is a real number, but in the %%numericallimitstut we also talked about different kinds of limit:

$\displaystyle \lim_{x \to \infty} f(x)$ #[and][y]# $\displaystyle \lim_{x \to -\infty} f(x)$.

Here, $x$ is allowed to approach $\pm \infty$; that is, to become arbitrarily large positive or negative. For instance, consider

$\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3)$.

The graph of $3x^2 - 5x - 3$ is a concave up parabola and the graphical approach to limits tells us that $\lim_{x \to \infty} (3x^2 - 5x - 3) = \infty$, as the graph rises without bound as $x$ becomes arbitrarily large. Thus,

$\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3) = \infty.$

#[Similarly,][De modo parecido,]#

$\displaystyle \lim_{x \to -\infty} (3x^2 - 5x - 3) = \infty$

#[because the graph also rises without bound as $x$ becomes arbitrarily large negative.][ya que la gráfica también sube sin límite mientras que $x$ se hace arbitrariamente grande negativo.]#

Algebraic argument
Write $3x^2 - 5x - 3$ as $\displaystyle x^2\left(3 - \frac{5}{x} - \frac{3}{x^2}\right)$ by factoring out the highest power of $x$.

#[Thus][Por lo tanto]#,

$\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3)$

$\displaystyle {}= \lim_{x \to \infty}$	$x^2\Bigl($	$3$	$-$	$\dfrac{5}{x}$	$-$	$\dfrac{3}{x^2}$	$\Bigr)$
		↓		↓		↓
		3	$ - $	0	$ - $	0

\\ \t

$\displaystyle = \lim_{x \to \infty} x^2(3) = \infty$

The terms $\dfrac{5}{x}$ and $\dfrac{3}{x^2}$ approach 0 because they have huge denominators as $x \to \infty$. In the language of forms, we can rewrite this argument as follows:

$\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3)$

\t $\displaystyle {}= \lim_{x \to \infty} x^2\left(3 - \frac{5}{x} - \frac{3}{x^2}\right)$ \\ \t $\displaystyle {}= \infty^2\left(3 - \frac{5}{\infty} - \frac{3}{\infty^2}\right)$ \\ \t $\displaystyle {}= \infty(3) = \infty$

where we have implicitly used the following rules of forms:

$\infty^2 = \infty \qquad$ \t Big × Big = Big \\ $\dfrac{k}{\infty} = 0$ \t $\dfrac{k}{\text{Big}}=0$ \\ $k(\infty) = \pm\infty \qquad$ \t k × Big = Big #[(the sign depends on the sign of $k$)][(el signo depende del signo de $k$).]#

Shortcut
Notice that all the terms in the original quadratic with powers less than the highest power (2) "disappear" in the end, meaning that we could have simply ignored them from the start:

$\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3)$ \t $\displaystyle {}= \lim_{x \to \infty} 3x^2$ \t #[Just ignore all but the highest power of $x$.][Simplemente ignora todo menos el poder más alto de $x$.]# \\ \t $\displaystyle {}= 3\infty^2 = \infty$ \t #[Rules for forms][Reglas para formas]#

The above analysis generalizes to limits at infinity of arbitrary polynomials, ratios of polynomials, and ratios of powers of polynomials:

Limits at ∞ of expressions with polynomials

Theorem P: Ignoring all but the highest power

When calculating the limit at infinity of any ratio of polynomials or constant powers of polynomials, we can ignore all but the terms with the highest power of $x$ in the polynomials that appear. (A proof can be found in Section 3.3 in Applied Calculus or Section 10.3 in Finite Mathematics and Applied Calculus.)

Examples

1. $\displaystyle \lim_{x \to \infty} (3x^2 - 5x - 3)$ \t $\displaystyle {}= \lim_{x \to \infty} 3x^2 = \infty$ \t #[See above][Ve arriba]# \\ 2. $\displaystyle \lim_{x \to \infty} \frac{x^2 - 1}{2x^2 - 4x}$ \t $\displaystyle {}= \lim_{x \to \infty} \frac{x^2}{2x^2} = \lim_{x \to \infty} \frac{1}{2} = \frac{1}{2}$ \\ 3. $\displaystyle \lim_{x \to \infty} \frac{-9x^3 - x^2 + 1}{2x^2 - 4x}$ \t $\displaystyle {}= \lim_{x \to \infty} \frac{-9x^3}{2x^2} = \lim_{x \to \infty} -9x = -\infty$ \t $-9(\infty) = -\infty$ \\ 4. $\displaystyle \lim_{x \to \infty} \frac{2x^2 - 4x}{-9x^3 - x^2 + 1}$ \t $\displaystyle {}= \lim_{x \to \infty} \frac{2x^2}{-9x^3} = \lim_{x \to \infty} -\frac{2}{9x} = 0$ \t $\dfrac{2}{\infty} = 0$

#[Note][Nota]# If you look at the first example above, notice that, when $x$ gets large, we have

$3(\text{Big}) - 5(\text{Big}) - 3$,

#[which we can think of as][Lo que podemos pensar como]#

\\ $\text{Big} - \text{Big} \qquad$ \t $3(\text{Big} = \text{Big}, 5(\text{Big}) = \text{Big}, 3((\text{Big}) - 3 = \text{Big}$

#[or][o]#

\\ $\infty - \infty$,

which, like 0/0, is an indeterminate form. Similarly, in the second example, after eliminating lower powers of $x$, but before canceling, we get another indeterminate form: $\dfrac{\infty}{\infty}$. We will look at these and other forms below.

Note that we Theorem P also applies to ratios of constant powers of polynomials, which including radicals:

More on determinate and indeterminate forms

#[Here is a list of the determinate and indeterminate forms we have looked at so far, as well as some new ones:][Aquí hay una lista de las formas determinadas e indeterminadas que hemos visto hasta ahora, así como algunas nuevas:]#

#[Some determinate forms][Algunas formas determinadas]#
#[Following are some determinate forms we have encountered in the above examples, together with a few additional ones:][Las siguientes son algunas formas determinadas que hemos encontrado en los ejemplos más arriba, juntos con algunas adicionales:]#

$\pm k \cdot \infty = \pm \infty \qquad$ \t $\pm$ #[Constant][Constante]# × #[Big][Grande]# = $\pm$ #[Big][Grande]# \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}(43x) = 43\infty =\infty$ \\ $k \pm \infty = \pm\infty$ \t #[Constant][Constante]# $\pm$ #[Big][Grande]# = $\pm$ #[Big][Grande]# \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}(43 - 2x) = 43 - 2\infty = 43 - \infty =-\infty$ \\ $\infty \cdot\infty =\infty$ \t #[Big][Grande]# × #[Big][Grande]# = #[Big][Grande]# \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to -\infty}(3x^2) = \lim_{x \to -\infty}(3x \cdot x) = (-\infty)(-\infty) = \infty$ \\ $\infty +\infty =\infty$ \t #[Big][Grande]# + #[Big][Grande]# = #[Big][Grande]# \t #[Example][Ejemplo]#:$\displaystyle \lim_{x \to \infty}(x^2 + 2x) = \infty^2 + 2\infty = \infty + \infty = \infty$ \\ $\pm \dfrac{k}{\infty} = 0$ \t $\displaystyle \pm \frac{\text{Constant}}{\text{Big}} = \text{Small}$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to -\infty}\frac{3}{6x^3} = 0$ \\ $\pm\dfrac{\infty}{k} = \pm\infty$ \t $\displaystyle \pm \frac{\text{Big}}{\text{Constant}} = \pm Big$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to -\infty}\frac{6x^3}{3} = \frac{6(-\infty)^3}{3} = \frac{-\infty}{3} = -\infty$ \\ $k^{\infty} = \infty$ #[if][si]# $k\gt 1.$ \t $(\text{Constant} \gt 1)^{Big} = \text{Big}$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}1.324^{x} = 1.324^{\infty} = \infty$ \\ $k^{\infty} = 0$ #[if][si]# $k\lt 1.$ \t $(\text{Constant} \lt 1)^{Big} = \text{Small}$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}0.9998^{x} = 0.9998^{\infty} = 0$ \\#[Reciprocal forms of the above two:][Formas recíprocas de las dos anteriores:]# \\ $k^{-\infty} = 0$ #[if][si]# $k\gt 1.$ \t $(\text{Constant} \gt 1)^{-Big} = \text{Small}$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}1.001^{-x} = 1.001^{-\infty} = 0$ \\ $k^{-\infty} = \infty$ #[if][si]# $k\lt 1.$ \t $(\text{Constant} \lt 1)^{-Big} = \text{Big}$ \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}0.9998^{-x} = 0.9998^{-\infty} = \infty$

#[Some indeterminate forms][Algunas formas indeterminadas]#
#[Remember: When we encounter these in a limit, we need to simplify or do some further analysis in order to determine whether the limit exists and its value when it does exist.][Recuerda: Cuando las encontramos en un límite, tenemos que simplificar o hacer más análisis para determinar si o no el límite existe y su valor cuando sí existe.]#

\\ $\dfrac{0}{0}$ \t $\displaystyle \frac{\text{Small}}{\text{Small}} = ???$ \t #[Enountered in Part A of this tutorial][Encontrada en Parte A de este tutorial]# \\ $\dfrac{\infty}{\infty}$ \t $\displaystyle \frac{\text{Big}}{\text{Big}} = ???$ \t #[Examples][Ejemplos]#: $\displaystyle \lim_{x \to -\infty}\frac{3x^2}{3x^2}$ #[or][o]# $\displaystyle \lim_{x \to \infty}\frac{-3x^2}{2x^4}$ \\ $\infty - \infty$ \t #[Big][Grande]# − #[Big][Grande]# = ??? \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to -\infty}(3x^2 - 4x)$ \\ $0\cdot\infty$ \t #[Small][Pequeño]# × #[Big][Grande]# = ??? \t #[Example][Ejemplo]#: $\displaystyle \lim_{x \to \infty}\frac{1}{x}(x^2 + 1)$

Now try the exercises in Section 10.3 in Finite Mathematics and Applied Calculus. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.

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