the behavior of a function as $x \to \infty$ and $x \to \text{-} \infty$

if $\lim_{x \to \infty} f (x) = L$ or $\lim_{x \to \text{-} \infty} f (x) = L,$ then $y = L$ is a horizontal asymptote of $f$

infinite limit at infinity

a function that becomes arbitrarily large as x becomes large

the limiting value, if it exists, of a function as $x \to \infty$ or $x \to \text{-} \infty$

the line $y = m x + b$ if $f (x)$ approaches it as $x \to \infty$ or $x \to \text{-} \infty$

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Calculus Volume 1 · 4.6 Limits at Infinity and Asymptotes

Limits at Infinity and Asymptotes: practice problems

30 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 251
For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.
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Exercises 253
For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.
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Exercises 255
For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.
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Exercises 257
For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.
$f (x) = \frac{x}{x - 2}, a = 2$
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Exercises 259
For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.
$f (x) = (x - 1)^{- 1 / 3}, a = 1$
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Exercises 261
For the following exercises, evaluate the limit.
$lim_{x \to \infty} \frac{1}{3 x + 6}$
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Exercises 263
For the following exercises, evaluate the limit.
$lim_{x \to \infty} \frac{x ^{2} - 2 x + 5}{x + 2}$
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Exercises 265
For the following exercises, evaluate the limit.
$lim_{x \to - \infty} \frac{x ^{4} - 4 x ^{3} + 1}{2 - 2 x ^{2} - 7 x ^{4}}$
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Exercises 267
For the following exercises, evaluate the limit.
$lim_{x \to - \infty} \frac{4 x ^{2} - 1}{x + 2}$
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Exercises 269
For the following exercises, evaluate the limit.
$lim_{x \to - \infty} \frac{4 x}{x ^{2} - 1}$
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Exercises 271
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = x - \frac{9}{x}$
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Exercises 273
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = \frac{x ^{3}}{4 - x ^{2}}$
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Exercises 275
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = sin (x) sin (2 x)$
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Exercises 277
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = \frac{x s i n ( x )}{x ^{2} - 1}$
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Exercises 279
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = \frac{1}{x ^{3} + x ^{2}}$
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Exercises 281
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = \frac{x ^{3} + 1}{x ^{3} - 1}$
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Exercises 283
For the following exercises, find the horizontal and vertical asymptotes.
$f (x) = x - sin x$
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Exercises 285
For the following exercises, construct a function $f (x)$ that has the given asymptotes.
$x = 1$ and $y = 2$
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Exercises 287
For the following exercises, construct a function $f (x)$ that has the given asymptotes.
$y = 4,$ $x = - 1$
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Exercises 289
For the following exercises, graph the function on a graphing calculator on the window $x = [- 5, 5]$ and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.
[T] $f (x) = \frac{1}{x + 10}$
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Exercises 291
For the following exercises, graph the function on a graphing calculator on the window $x = [- 5, 5]$ and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.
[T] $lim_{x \to - \infty} x^{2} + 10 x + 25$
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Exercises 293
For the following exercises, graph the function on a graphing calculator on the window $x = [- 5, 5]$ and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.
[T] $lim_{x \to \infty} \frac{3 x + 2}{x + 5}$
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Exercises 295
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = x^{3} - 3 x^{2} + 4$
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Exercises 297
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = \frac{x ^{3} + 4 x ^{2} + 3 x}{3 x + 9}$
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Exercises 299
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = x^{2} - 5 x + 4$
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Exercises 301
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = \frac{c o s x}{x},$ on $x = [- 2 π, 2 π]$
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Exercises 303
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = x tan x, x = [- π, π]$
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Exercises 305
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
$y = x^{2} sin (x), x = [- 2 π, 2 π]$
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Exercises 307
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
For $f (x) = \frac{P ( x )}{Q ( x )}$ to have an asymptote at $x = 0,$ then the polynomials $P (x)$ and must have what relation?
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Exercises 309
For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.
Both $f (x) = \frac{1}{( x - 1 )}$ and $g (x) = \frac{1}{( x - 1 ) ^{2}}$ have asymptotes at and What is the most obvious difference between these two functions?
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Key terms · 5

end behavior: the behavior of a function as $x \to \infty$ and $x \to \text{-} \infty$
horizontal asymptote: if $\lim_{x \to \infty} f (x) = L$ or $\lim_{x \to \text{-} \infty} f (x) = L,$ then $y = L$ is a horizontal asymptote of $f$
infinite limit at infinity: a function that becomes arbitrarily large as x becomes large
limit at infinity: the limiting value, if it exists, of a function as $x \to \infty$ or $x \to \text{-} \infty$
oblique asymptote: the line $y = m x + b$ if $f (x)$ approaches it as $x \to \infty$ or $x \to \text{-} \infty$

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Limits at Infinity and Asymptotes: practice problems

30 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 251

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.

Exercises 253

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = −1 from the left, and on the other side of x = −1, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.

Exercises 255

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.

Exercises 257

For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.

$f (x) = \frac{x}{x - 2}, a = 2$

Exercises 259

For the following functions $f (x),$ determine whether there is an asymptote at $x = a .$ Justify your answer without graphing on a calculator.

$f (x) = (x - 1)^{- 1 / 3}, a = 1$

Exercises 261

For the following exercises, evaluate the limit.

$lim_{x \to \infty} \frac{1}{3 x + 6}$

Exercises 263

For the following exercises, evaluate the limit.

$lim_{x \to \infty} \frac{x ^{2} - 2 x + 5}{x + 2}$

Exercises 265

For the following exercises, evaluate the limit.

$lim_{x \to - \infty} \frac{x ^{4} - 4 x ^{3} + 1}{2 - 2 x ^{2} - 7 x ^{4}}$

Exercises 267

For the following exercises, evaluate the limit.

$lim_{x \to - \infty} \frac{4 x ^{2} - 1}{x + 2}$

Exercises 269

For the following exercises, evaluate the limit.

$lim_{x \to - \infty} \frac{4 x}{x ^{2} - 1}$

Exercises 271