# 2.2 The limit of a function  (Page 7/12)

 Page 7 / 12

## Key concepts

• A table of values or graph may be used to estimate a limit.
• If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
• If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
• We may use limits to describe infinite behavior of a function at a point.

## Key equations

• Intuitive Definition of the Limit
$\underset{x\to a}{\text{lim}}f\left(x\right)=L$
• Two Important Limits
$\underset{x\to a}{\text{lim}}x=a\phantom{\rule{1em}{0ex}}\underset{x\to a}{\text{lim}}c=c$
• One-Sided Limits
$\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=L\phantom{\rule{1em}{0ex}}\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=L$
• Infinite Limits from the Left
$\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=\text{+}\infty \phantom{\rule{1em}{0ex}}\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty$
• Infinite Limits from the Right
$\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=\text{+}\infty \phantom{\rule{1em}{0ex}}\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty$
• Two-Sided Infinite Limits
$\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty :\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=\text{+}\infty$ and $\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=\text{+}\infty$
$\underset{x\to a}{\text{lim}}f\left(x\right)=\text{−}\infty :\underset{x\to {a}^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty$ and $\underset{x\to {a}^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty$

For the following exercises, consider the function $f\left(x\right)=\frac{{x}^{2}-1}{|x-1|}.$

[T] Complete the following table for the function. Round your solutions to four decimal places.

x $f\left(x\right)$ x $f\left(x\right)$
0.9 a. 1.1 e.
0.99 b. 1.01 f.
0.999 c. 1.001 g.
0.9999 d. 1.0001 h.

What do your results in the preceding exercise indicate about the two-sided limit $\underset{x\to 1}{\text{lim}}f\left(x\right)?$ Explain your response.

$\underset{x\to 1}{\text{lim}}f\left(x\right)$ does not exist because $\underset{x\to {1}^{-}}{\text{lim}}f\left(x\right)=-2\ne \underset{x\to {1}^{+}}{\text{lim}}f\left(x\right)=2.$

For the following exercises, consider the function $f\left(x\right)={\left(1+x\right)}^{1\text{/}x}.$

[T] Make a table showing the values of f for $x=-0.01,-0.001,-0.0001,-0.00001$ and for $x=0.01,0.001,0.0001,0.00001.$ Round your solutions to five decimal places.

x $f\left(x\right)$ x $f\left(x\right)$
−0.01 a. 0.01 e.
−0.001 b. 0.001 f.
−0.0001 c. 0.0001 g.
−0.00001 d. 0.00001 h.

What does the table of values in the preceding exercise indicate about the function $f\left(x\right)={\left(1+x\right)}^{1\text{/}x}?$

$\underset{x\to 0}{\text{lim}}{\left(1+x\right)}^{1\text{/}x}=2.7183$

To which mathematical constant does the limit in the preceding exercise appear to be getting closer?

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

[T] $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}2x}{x};\phantom{\rule{0.2em}{0ex}}±0.1,±0.01,±0.001,±.0001$

x $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}2x}{x}$ x $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}2x}{x}$
−0.1 a. 0.1 e.
−0.01 b. 0.01 f.
−0.001 c. 0.001 g.
−0.0001 d. 0.0001 h.

a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}2x}{x}=2$

[T] $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}3x}{x}$ ±0.1, ±0.01, ±0.001, ±0.0001

X $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}3x}{x}$ x $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}3x}{x}$
−0.1 a. 0.1 e.
−0.01 b. 0.01 f.
−0.001 c. 0.001 g.
−0.0001 d. 0.0001 h.

Use the preceding two exercises to conjecture (guess) the value of the following limit: $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}ax}{x}$ for a , a positive real value.

$\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}ax}{x}=a$

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

$\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{{x}^{2}+x-6}$

x $\frac{{x}^{2}-4}{{x}^{2}+x-6}$ x $\frac{{x}^{2}-4}{{x}^{2}+x-6}$
1.9 a. 2.1 e.
1.99 b. 2.01 f.
1.999 c. 2.001 g.
1.9999 d. 2.0001 h.

$\underset{x\to 1}{\text{lim}}\left(1-2x\right)$

x $1-2x$ x $1-2x$
0.9 a. 1.1 e.
0.99 b. 1.01 f.
0.999 c. 1.001 g.
0.9999 d. 1.0001 h.

a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000; $\underset{x\to 1}{\text{lim}}\left(1-2x\right)=-1$

$\underset{x\to 0}{\text{lim}}\frac{5}{1-{e}^{1\text{/}x}}$

x $\frac{5}{1-{e}^{1\text{/}x}}$ x $\frac{5}{1-{e}^{1\text{/}x}}$
−0.1 a. 0.1 e.
−0.01 b. 0.01 f.
−0.001 c. 0.001 g.
−0.0001 d. 0.0001 h.

$\underset{z\to 0}{\text{lim}}\frac{z-1}{{z}^{2}\left(z+3\right)}$

z $\frac{z-1}{{z}^{2}\left(z+3\right)}$ z $\frac{z-1}{{z}^{2}\left(z+3\right)}$
−0.1 a. 0.1 e.
−0.01 b. 0.01 f.
−0.001 c. 0.001 g.
−0.0001 d. 0.0001 h.

a. −37.931934; b. −3377.9264; c. −333,777.93; d. −33,337,778; e. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889 $\underset{x\to 0}{\text{lim}}\frac{z-1}{{z}^{2}\left(z+3\right)}=\text{−}\infty$

What is a independent variable
a variable that does not depend on another.
Andrew
solve number one step by step
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
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how to prove sin x - sin x cos^2 x=sin^3x?
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
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Alif
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Salim
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what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
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a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
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what is hyperbolic function
find volume of solid about y axis and y=x^3, x=0,y=1
3 pi/5
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what is the power rule
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how do i deal with infinity in limits?
f(x)=7x-x g(x)=5-x
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5x-5
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what is domain
difference btwn domain co- domain and range
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x
Verna
The set of inputs of a function. x goes in the function, y comes out.
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where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
pls solve it i Want to see the answer
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ok
Friendz
differentiate each term
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why do we need to study functions?
to understand how to model one variable as a direct relationship to another variable
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