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Standard Level

11.02 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
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
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
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
11

Consider the graph of f(x):

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

-2
-1
1
2
3
4
5
6
7
8
9
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
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
2
3
4
5
6
7
8
9
x
20
40
60
80
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
2
3
4
x
1
2
3
4
5
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
4
5
6
x
-2
-1
1
2
3
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
2
3
4
5
6
x
-3
-2
-1
1
2
3
4
5
6
7
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
2
3
4
5
6
x
1
2
3
4
5
6
7
8
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
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
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