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8.02 Introduction to limits

Worksheet
Tables
1

Consider the function f \left( x \right) = \dfrac{1}{7 - x}.

a

Complete the first table of values, in which x \lt 7.

b

Complete the second table of values, in which x \gt 7.

c

Find the limit of f \left( x \right) as the value of x approaches 7.

x566.96.99
f(x)
x987.17.01
f(x)
2

Consider the function f \left( x \right) = 5 x^{2} + 1:

a

Complete the table to find the exact values of f \left( x \right) as x gets closer and closer to 2 from the left, and closer and closer to 2 from the right:

x1.91.991.9992.0012.012.1
f(x)
b

Find the value of \lim_{x \to 2}\left( 5 x^{2} + 1\right).

3

Consider the function f \left( x \right) = \dfrac{2 - x}{x^{2} + 2}:

a

Complete the table to find the values of f \left( x \right) as x gets closer and closer to 0 from the left, and closer and closer to 0 from the right. Round your answers to four decimal places.

x-0.1-0.01-0.0010.0010.010.1
f(x)
b

Find the value of \lim_{x \to 0}\left(\dfrac{2 - x}{x^{2} + 2}\right).

4

Consider the function f \left( x \right) = \dfrac{x^{2} - 4 x}{x - 4}.

a

Complete the table to find the values of f \left( x \right) as x gets closer and closer to 4 from the left, and closer and closer to 4 from the right:

x3.93.993.9994.0014.014.1
f \left( x \right)
b

Find the value of \lim_{x \to 4}\left(\dfrac{x^{2} - 4 x}{x - 4}\right).

5

Consider the function f \left( x \right) = \dfrac{x^{3} + x + 2}{x + 1}.

a

Complete the following table:

x-1.1-1.01-1.001-0.999-0.99-0.9
f \left( x \right)
b

Find \lim_{x \to - 1 } f \left( x \right).

6

Consider the function f \left( x \right) = \dfrac{\sqrt{x} + 4}{x - 5} :

a

Complete the following table, rounding all values to two decimal places:

x4.94.994.9995.0015.015.1
f \left( x \right)
b

Does \lim_{x \to 5} f \left( x \right) exist? Explain your answer.

7

Consider the limit: \lim_{x \to - 5 }\left(\dfrac{x^{2} + 4}{x + 5}\right).

a

Complete the table.

x-5.1-5.01-5.001-5-4.999-4.99-4.9
\dfrac{x^2+4}{x+5}-
b

Does the above limit exist? Explain your answer.

8

Consider the limit: \lim_{x \to 3}\left(\dfrac{e^{x - 3} + x - 4}{x - 3}\right).

a

Complete the following table:

x2.92.992.99933.0013.013.1
\left(\dfrac{e^{x - 3} + x - 4}{x - 3}\right)-
b

Does the above limit exist? Explain your answer.

Graphs
9

Consider the graph of the piecewise function g \left( x \right):

a

If we start at - 4 and move along g \left( x \right) to the right towards x = - 2, what y-value do we approach?

b

If we start at 0 and move along g \left( x \right) to the left towards x = - 2, what y-value do we approach?

c

Does the \lim_{x \to -2} g(x) exist? Explain your answer.

-5
-4
-3
-2
-1
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x
-5
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y
10

Consider the graph of the function f\left(x\right) = \dfrac{x^{2} - 1}{x + 1}:

a

If we start at x = - 3 and move along the function to the right towards x = -1, what y-value do we approach?

b

If we start at x = 1 and move along the function to the left towards x = -1, what y-value do we approach?

c

Describe the behaviour of \\ f\left(x\right) = \dfrac{x^{2} - 1}{x + 1} as x approaches -1.

d

Write part (c) using limit notation.

-4
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11

Consider the graph of f(x):

Does the limit \lim_{x \to 3} f \left( x \right) exist? Explain your answer.

-2
-1
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x
-9
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1
y
12

Consider the graph of f(x):

Does the limit \lim_{x \to 0} f \left( x \right) exist? Explain your answer.

-3
-2
-1
1
2
3
x
-4
-3
-2
-1
1
2
y
13

Consider the graph of y = \dfrac{x + 3}{\left(x - 5\right)^{2}}:

Does the limit \lim_{x \to 5}\left(\dfrac{x + 3}{\left(x - 5\right)^{2}}\right) exist? Explain your answer.

-1
1
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x
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y
14

Consider the graph of y = \dfrac{x^{2}}{1 - \cos x}:

Does the limit \lim_{x \to 0}\left(\dfrac{x^{2}}{1 - \cos x}\right) exist? Explain your answer.

-4
-3
-2
-1
1
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x
1
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y
15

Consider the graph of the function f \left( x \right):

Find the value of \lim_{x \to 3} f \left( x \right).

1
2
3
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5
6
x
-2
-1
1
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4
y
16

Consider the graph of the function f \left( x \right):

Find the value of \lim_{x \to 0} f \left( x \right).

-1
1
x
-2
-1
1
2
y
17

Consider the graph of the function f \left( x \right):

Find the value of \lim_{x \to 3} f \left( x \right).

-3
-2
-1
1
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x
-3
-2
-1
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y
18

Consider the graph of the function f \left( x \right):

Find the value of \lim_{x \to 3} f \left( x \right).

1
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1
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y
19

Consider the graph of the function f \left( x \right):

Find the value of \lim_{x \to 2} f \left( x \right).

1
2
3
x
1
2
3
y
20

Consider the graph of the function \\ f \left( x \right) = \dfrac{1}{x + 4}:

Find the value of:

a
\lim_{x \to - 4 ^+}\left(\dfrac{1}{x + 4}\right)
b
\lim_{x \to - 4 ^-}\left(\dfrac{1}{x + 4}\right)
-8
-6
-4
-2
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x
-8
-6
-4
-2
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y
Limits
21

Consider the function f \left( x \right) = \dfrac{1}{x + 3}:

a

Find the value of \dfrac{1}{x + 3} at x = - 3.

b

Find the value of \lim_{x \to - 3 }\left(\dfrac{1}{x + 3}\right).

22

Consider the function f \left( x \right) = \dfrac{x^{2} + 7 x + 10}{x + 5} :

a

Find the value of f\left(x\right) at x = - 5.

b

Find the value of \lim_{x \to - 5 } f\left(x\right).

23

Find the value of the following limits:

a

\lim_{x \to 5} 9

b

\lim_{x \to 3}\left( - 2 \right)

c

\lim_{x \to 6}\left( 5 x\right)

d

\lim_{x \to 6}\left( 4 x^{2}\right)

e

\lim_{x \to 0}\left(\dfrac{x^{2} - 6 x}{x}\right)

f

\lim_{x \to 1}\left( 3 x^{4} - 5 x^{3} + 2\right)

g

\lim_{x \to 5} \sqrt{41 - x}

h

\lim_{x \to - 3 }\left(\dfrac{2 x + 5}{5 x + 2}\right)

i

\lim_{x \to 0}\left(\dfrac{x^{2} - 25}{x^{2} + 25}\right)

j

\lim_{x \to 3}\left(\dfrac{x^{2} - 9 x + 18}{x - 3}\right)

k

\lim_{x \to - 4 }\left(\dfrac{x^{2} + 6 x + 8}{x + 4}\right)

l

\lim_{x \to - 2 }\left(\dfrac{x^{2} + 5 x + 6}{x + 2}\right)

m

\lim_{x \to 3}\left(\dfrac{x^{2} - x - 6}{x^{2} - 9}\right)

n

\lim_{x \to - 2 }\left(\dfrac{x^{2} - 4}{x + 2}\right)

o

\lim_{x \to - 9 }\left(\dfrac{x^{2} + 9 x}{x^{2} - 81}\right)

p

\lim_{x \to 3}\left(\dfrac{\left(x - 3\right)^{2}}{x^{2} - 8 x + 15}\right)

q

\lim_{x \to - 4 }\left(\dfrac{\left(x + 4\right)^{2}}{5 x^{2} + 24 x + 16}\right)

Limits to infinity
24

Consider the function f \left( x \right) = \dfrac{3}{x + 2}.

a

What value does the function approach as x approaches infinity?

b

What value does the function approach as x approaches negative infinity?

c

What value does the function approach as x approaches zero?

25

Consider the function f \left( x \right) = 3^{x} :

a

Find the limit of the function as x approaches infinity.

b

Find the limit of the function as x approaches negative infinity.

c

Find the limit of the function as x approaches zero.

26

Consider the graph of the function:

a

What value does y approach as x approaches infinity?

b

What value does y approach as x approaches negative infinity?

c

What value does y approach as x approaches zero?

-5
5
x
-5
5
y
27

Consider the graph of the function:

a

Find the limit of y as x approaches infinity.

b

Find the limit of y as x approaches negative infinity.

c

Find the limit of y as x approaches zero.

-5
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-5
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Applications
28

Use limit notation to represent the following statement:

The value that the function y = x + 4 approaches as x approaches 168

29

The price P \left( t \right) of coffee in a cafe changes as a function of time, where t is in years.

If P(t)=4 \times 1.025^t, find the amount that the price of coffee approaches after 13 years.

30

A ball is dropped, and on the first bounce it reaches a height of 2 \text{ m} and after each bounce it reaches one quarter of its previous height. On the nth bounce, it travels a distance of d \left( n \right).

Write an expression for d(n) and find the value that the distance travelled by the ball on each bounce approaches as the number of bounces n goes on forever.

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Outcomes

2.4.1.3

examine the behaviour of the difference quotient [𝑓(𝑥+ℎ)−𝑓(𝑥)]/h ℎ as ℎ→0 as an informal introduction to the concept of a limit

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