LemonmathSelf-study sheet

Grade 12 · Lesson 1.3

Asymptotes

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Goals

Understand and find the horizontal and vertical asymptotes of a graph with limits.
Find an oblique asymptote by polynomial division or with the limit formulas.
Find the asymptotes of $y=\dfrac{ax+b}{cx+d}$ and $y=\dfrac{ax^2+bx+c}{mx+n}$ , and count the asymptotes of a graph.
Explain what an asymptote means in a real-life problem.

Remember

Horizontal asymptote $y=y_0$ : $\lim\limits_{x\to+\infty}f=y_0$ or $\lim\limits_{x\to-\infty}f=y_0$ . Compute both sides.
Vertical asymptote $x=x_0$ : a one-sided limit at $x_0$ is $\pm\infty$ . For a rational function: find the roots of the denominator, then check the limit.
Oblique asymptote $y=ax+b$ : $a=\lim\dfrac{f(x)}{x}$ , $b=\lim[f(x)-ax]$ ; for $\dfrac{ax^2+bx+c}{mx+n}$ , divide and drop the remainder.
$y=\dfrac{ax+b}{cx+d}$ : vertical asymptote $x=-\dfrac dc$ , horizontal asymptote $y=\dfrac ac$ .
In a figure, asymptotes are dashed lines; the graph gets close to them but need not cross or touch them.

Exercises

Exercise 1

Find the asymptotes of the graph of $y=\dfrac{3x+2}{x-4}$ .

Exercise 2

Find the asymptotes of the graph of $y=\dfrac{1-2x}{x+3}$ .

Exercise 3

Find the asymptotes of the graph of $y=\dfrac{x^2+2x+3}{x+1}$ .

Exercise 4

How many asymptotes does the graph of $y=\dfrac{x^2-4}{x^2-3x+2}$ have? Which lines are they?

Exercise 5

Consider $y=\dfrac{x^2-3x+4}{x-2}$ . Is each claim true or false?

a) The line $x=2$ is a vertical asymptote of the graph.

b) The graph has no horizontal asymptote.

c) The line $y=x+1$ is an oblique asymptote of the graph.

d) The graph has exactly two asymptotes.

Exercise 6

Find the total number of vertical and horizontal asymptotes of the graph of $y=\dfrac{2x+1}{x^2-4}$ .

Exercise 7

A tank holds $200$ litres of clean water. Salt water with a concentration of $30$ grams per litre is pumped in at $10$ litres per minute. After $t$ minutes, the salt concentration in the tank is $C(t)=\dfrac{30t}{20+t}$ (grams per litre). Find the horizontal asymptote of the graph of $C(t)$ and explain what it means.

Exercise 8 Review

Compute $\lim\limits_{x\to+\infty}\left(\sqrt{x^2+2x}-x\right)$ , and deduce a horizontal asymptote of the graph of $y=\sqrt{x^2+2x}-x$ .

Exercise 9 Review

Prove that the graph of $y=\dfrac{2x^2-1}{x^2-2x+3}$ has no vertical asymptote, then find its horizontal asymptote.

Exercise 10 Challenge

Find the oblique asymptotes of the graph of $y=\sqrt{x^2+4x}$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The vertical asymptote of the graph of

y=\dfrac{3x-1}{x-2}

is the line

x=-2

.B.

x=2

.C.

y=3

.D.

x=\dfrac13

Question 2. The horizontal asymptote of the graph of

y=\dfrac{1+6x}{3-2x}

is the line

x=\dfrac32

.B.

y=3

.C.

y=\dfrac13

.D.

y=-3

Question 3. The oblique asymptote of the graph of

y=\dfrac{x^2+3x-2}{x+1}

is the line

y=x+2

.B.

y=x+3

.C.

y=x-2

.D.

y=x

Question 4. A function

y=f(x)

has

\lim\limits_{x\to+\infty}f(x)=2

and

\lim\limits_{x\to-\infty}f(x)=-2

. Which statement is correct?

A. The graph has exactly one horizontal asymptote.B. The graph has two vertical asymptotes,

x=2

and

x=-2

.C. The graph has two horizontal asymptotes,

y=2

and

y=-2

.D. The graph has no horizontal asymptote.

Part II. True or false

Question 1. Let

y=\dfrac{2x^2-3x+1}{x-2}

a) The line $x=2$ is a vertical asymptote of the graph.	T / F
b) The line $y=2$ is a horizontal asymptote of the graph.	T / F
c) The line $y=2x+1$ is an oblique asymptote of the graph.	T / F
d) The oblique asymptote and the two coordinate axes form a triangle of area $\dfrac14$ .	T / F

Part III. Short answer

Question 1. How many vertical and horizontal asymptotes does the graph of

y=\dfrac{\sqrt{x+3}-2}{x^2-1}

have in total?

Question 2. The concentration of a substance in a water tank after

t

minutes is

C(t)=\dfrac{25t}{3t+8}

(mg/l). The graph of

C(t)

has the horizontal asymptote

C=c

. Find

c

, giving your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $y=\dfrac{ax+b}{cx+d}$ has the vertical asymptote $x=-\dfrac dc$ and the horizontal asymptote $y=\dfrac ac$ .

Steps:

Compute the limits as $x\to\pm\infty$ .
Find the root of the denominator and check that the numerator is not $0$ there.

Exercise 2.

What you need: the coefficient of $x$ in the numerator is $-2$ ; watch the order of the terms.

Steps:

Write the numerator as $-2x+1$ to read off $a=-2$ .
Find the root of the denominator.

Exercise 3.

What you need: polynomial division: if $y=px+q+\dfrac{r}{x+1}$ , the oblique asymptote is $y=px+q$ .

Steps:

Divide $x^2+2x+3$ by $x+1$ .
Find the vertical asymptote from the denominator.

Exercise 4.

What you need: if a root of the denominator is also a root of the numerator, the fraction can be simplified, and there may be no vertical asymptote there.

Steps:

Factorise the numerator and the denominator, and simplify.
Check the limit at each root of the denominator and at infinity.

Exercise 5.

What you need: divide to write $y=px+q+\dfrac{r}{x-2}$ .

Steps:

Divide $x^2-3x+4$ by $x-2$ .
Check each claim.

Exercise 6.

What you need: if the degree of the numerator is less than the degree of the denominator, the limit at infinity is $0$ .

Steps:

Find the roots of the denominator and check the numerator there.
Compute the limits at $\pm\infty$ .

Exercise 7.

What you need: the amount of salt is $30\cdot10t=300t$ grams and the volume is $200+10t$ litres; the horizontal asymptote is the limit as $t\to+\infty$ .

Steps:

Check the formula: $\dfrac{300t}{200+10t}=\dfrac{30t}{20+t}$ .
Compute $\lim\limits_{t\to+\infty}C(t)$ .

Exercise 8.

What you need: multiply and divide by the conjugate $\sqrt{x^2+2x}+x$ to remove the form $\infty-\infty$ (limits of functions, grade 11).

Steps:

Write $\sqrt{x^2+2x}-x=\dfrac{2x}{\sqrt{x^2+2x}+x}$ .
Divide the numerator and the denominator by $x$ (with $x>0$ ).

Exercise 9.

What you need: a quadratic expression with $\Delta<0$ has no roots (see the sign of a quadratic expression).

Steps:

Compute the discriminant of the denominator.
Divide the numerator and the denominator by $x^2$ to find the limit at infinity.

Exercise 10.

What you need: $a=\lim\dfrac{f(x)}{x}$ , $b=\lim[f(x)-ax]$ , computed separately as $x\to+\infty$ and as $x\to-\infty$ ; when $x<0$ , $\sqrt{x^2}=-x$ .

Steps:

Find the domain.
Compute $a$ and $b$ as $x\to+\infty$ (use the conjugate).
Repeat as $x\to-\infty$ .

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