In the previous section, we saw two examples which demonstrated the basics of limits. In this section, we will see a few more examples.
Example 3:
This can be written in 3 steps:
1. Fig.13.3 below shows the graph of the function y = f(x) = x2
 |
| Fig.13.3 |
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.5 below:
Table 13.5
$\begin{array}{cc}
{}&{\color{green}x}
& {-2}&
{\color{green}{-1}}& {-0.5}& {\color{green}{-0.1}}&
{-0.01}& {\color{green}{-0.001}}& {-0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{4}& {\color{green}{1}}& {0.25}& {\color{green}{0.01}}&
{0.0001}& {\color{green}{10^{-6}}}& {10^{-8}}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the left, f(x) becomes closer and closer to 0
•
We can write:
$$\lim_{x\rightarrow 0^{-}} f(x) = 0$$
3. Let us find the limit of f(x) as x approaches 0 from the right. For that, we prepare table 13.6 below:
Table 13.6
{}&{\color{green}x}
& {2}& {\color{green}{1}}& {0.5}& {\color{green}{0.1}}& {0.01}& {\color{green}{0.001}}& {0.0001}
&{} \\
{}&{\color{green}{f(x)}}
& {4}& {\color{green}{1}}& {0.25}& {\color{green}{0.01}}& {0.0001}& {\color{green}{10^{-6}}}& {10^{-8}}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the right, f(x) becomes closer and closer to 0
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 0$$
• In this example, we see that:
♦ Left side limit of f(x) at 0 is 0
♦ Right side limit of f(x) at 0 is 0
♦ Value of f(x) at 0 is also 0
✰ That means, limit at 0 can be obtained by putting x = 0.
✰ f(x) = x2
✰ f(0) = 02 = 0
Example 4
This can be written in 3 steps:
1. Fig.13.4 below shows the graph of the function: f(x) = 3
 |
| Fig.13.4 |
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.7 below:
Table 13.7
$\begin{array}{cc}
{}&{\color{green}x}
& {-2}&
{\color{green}{-1}}& {-0.5}& {\color{green}{-0.1}}&
{-0.01}& {\color{green}{-0.001}}& {-0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{3}& {\color{green}{3}}& {3}& {\color{green}{3}}&
{3}& {\color{green}{3}}& {3}
&{} \\
\end{array}$
•
It is clear that:
♦ x become closer and closer to 0 from the left
♦ f(x) is 3 for all those increasing values of x
•
We can write:
$$\lim_{x\rightarrow 0^{-}} f(x) = 3$$
◼ In fact, we can write:
$$\lim_{x\rightarrow a^{-}} f(x) = 3$$
♦ Where 'a' is any real number
•
The reader may think that:
♦ f(x) is not approaching '3'.
♦ '3' is already present.
•
This thought is correct.
Even then, we can consider 3 as the limit because, as x approaches 0, there is no value other than '3'. Also, the value '3' will not be exceeded.
3. Let us find the limit of f(x) as x approaches 0 from the right. For that, we prepare table 13.8 below:
Table 13.8
$\begin{array}{cc}
{}&{\color{green}x}
& {2}&
{\color{green}{1}}& {0.5}& {\color{green}{0.1}}&
{0.01}& {\color{green}{0.001}}& {0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{3}& {\color{green}{3}}& {3}& {\color{green}{3}}&
{3}& {\color{green}{3}}& {3}
&{} \\
\end{array}$
•
It is clear that:
♦ x become closer and closer to 0 from the right
♦ f(x) is 3 for all those decreasing values of x
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 3$$
◼ In fact, we can write:
$$\lim_{x\rightarrow a^{+}} f(x) = 3$$
♦ Where 'a' is any real number
• In this example, we see that:
♦ Left side limit of f(x) at 0 is 3
♦ Right side limit of f(x) at 0 is 3
♦ Value of f(x) at 0 is also 3
✰ That means, limit at 0 can be obtained by putting x = 0.
✰ f(x) = 3
✰ f(0) = 3
Example 5
This can be written in 3 steps:
1. Fig.13.5 below shows the graph of the function:
$$f(x) =
\begin{cases}
1, & {x \le 0} \\[2ex]
2, & {x > 0}
\end{cases}$$
 |
| Fig.13.5 |
•
Note that:
♦ the point (0,1) is marked with a filled circle.
♦ the point (0,2) is marked with an ordinary circle.
•
We saw the significance of such markings in fig.13.2 of the previous section.
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.9 below:
Table 13.9
$\begin{array}{cc}
{}&{\color{green}x}
& {-2}&
{\color{green}{-1}}& {-0.5}& {\color{green}{-0.1}}&
{-0.01}& {\color{green}{-0.001}}& {-0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{1}& {\color{green}{1}}& {1}& {\color{green}{1}}&
{1}& {\color{green}{1}}& {1}
&{} \\
\end{array}$
•
It is clear that:
♦ x become closer and closer to 0 from the left
♦ f(x) is 1 for all those increasing values of x
•
We can write:
$$\lim_{x\rightarrow 0^{-}} f(x) =1$$
3. Let us find the limit of f(x) as x approaches 0 from the right. For that, we prepare table 13.10 below:
Table 13.10
$\begin{array}{cc}
{}&{\color{green}x}
& {2}&
{\color{green}{1}}& {0.5}& {\color{green}{0.1}}&
{0.01}& {\color{green}{0.001}}& {0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{2}& {\color{green}{2}}& {2}& {\color{green}{2}}&
{2}& {\color{green}{2}}& {2}
&{} \\
\end{array}$
•
It is clear that:
♦ x become closer and closer to 0 from the right
♦ f(x) is 2 for all those decreasing values of x
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 2$$
• In this example, we see that:
♦ Left side limit of f(x) at 0 is 1
♦ Right side limit of f(x) at 0 is 2
♦ Left side and right side limits are different.
♦ In such cases, we say that: limit does not exist.
•
In this example,
♦ The limit as x approaches zero does not exist.
♦ But other limits exist.
✰ For example: the limit as x approaches 5 is 2
✰ Another example: the limit as x approaches -7 is 1
Example 6
This can be written in 3 steps:
1. Fig.13.6 below shows the graph of the function:
$$f(x) =
\begin{cases}
x-2, & {x < 0} \\[2ex]
0, & {x = 0} \\[2ex]
x+2, & {x > 0}
\end{cases}$$
 |
| Fig.13.6 |
•
Note that:
♦ the point (0,2) is marked with an ordinary circle.
♦ the point (0,-2) is also marked with an ordinary circle.
•
We saw the significance of such markings in fig.13.2 of the previous section.
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.11 below:
Table 13.11
$\begin{array}{cc}
{}&{\color{green}x}
& {-2}&
{\color{green}{-1}}& {-0.5}& {\color{green}{-0.1}}&
{-0.01}& {\color{green}{-0.001}}& {-0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{-4}& {\color{green}{-3}}& {-2.5}& {\color{green}{-2.1}}&
{-2.01}& {\color{green}{-2.001}}& {-2.0001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the left, f(x) becomes closer and closer to -2
•
We can write:
$$\lim_{x\rightarrow 0^{-}} f(x) = -2$$
3. Let us find the limit of f(x) as x approaches 0 from the right. For that, we prepare table 13.12 below:
Table 13.12
$\begin{array}{cc}{}&{\color{green}x}
& {2}& {\color{green}{1}}& {0.5}& {\color{green}{0.1}}& {0.01}& {\color{green}{0.001}}& {0.0001}
&{} \\
{}&{\color{green}{f(x)}}
& {4}& {\color{green}{3}}& {2.5}& {\color{green}{2.1}}& {2.01}& {\color{green}{2.001}}& {2.0001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the right, f(x) becomes closer and closer to 2
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 2$$
• In this example, we see that:
♦ Left side limit of f(x) at 0 is -2
♦ Right side limit of f(x) at 0 is 2
♦ Left side and right side limits are different.
♦ In such cases, we say that: limit does not exist
•
In this example,
♦ The limit as x approaches zero does not exist.
♦ But other limits exist.
✰ For example: the limit as x approaches 5 is 7
✰ Another example: the limit as x approaches -8 is -10
Example 7
This can be written in 3 steps:
1. Fig.13.7 below shows the graph of the function:
$$f(x) =
\begin{cases}
x+2, & {x \ne 1} \\[2ex]
0, & {x = 1}
\end{cases}$$
 |
| Fig.13.7 |
•
Note that:
♦ the point (1,0) is marked with a filled circle.
♦ the point (1,3) is marked with an ordinary circle.
•
We saw the significance of such markings in fig.13.2 of the previous section.
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.13 below:
Table 13.13
$\begin{array}{cc}
{}&{\color{green}x}
& {-1}&
{\color{green}{0.5}}& {0}& {\color{green}{0.5}}&
{0.9}& {\color{green}{0.99}}& {0.999}
&{} \\
{}&{\color{green}{f(x)}}
&
{1}& {\color{green}{1.5}}& {2}& {\color{green}{2.5}}&
{2.9}& {\color{green}{2.99}}& {2.999}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the left, f(x) becomes closer and closer to 3
•
We can write:
$$\lim_{x\rightarrow 1^{-}} f(x) = 3$$
3. Let us find the limit of f(x) as x approaches 1 from the right. For that, we prepare table 13.14 below:
Table 13.14
$\begin{array}{cc}{}&{\color{green}x}
& {3}& {\color{green}{2}}& {1.5}& {\color{green}{1.1}}& {1.01}& {\color{green}{1.001}}& {1.0001}
&{} \\
{}&{\color{green}{f(x)}}
& {5}& {\color{green}{4}}& {3.5}& {\color{green}{3.1}}& {3.01}& {\color{green}{3.001}}& {3.0001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the right, f(x) becomes closer and closer to 3
•
We can write:
$$\lim_{x\rightarrow 1^{+}} f(x) = 3$$
• In this example, we see that:
♦ Left side limit of f(x) at 1 is 3
♦ Right side limit of f(x) at 1 is 3
♦ Value of f(x) at 1 is not 3
In the next section, we will see a few more examples.
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