Chapter 13.1 - When Limit At A Point Does Not Exist

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 

$\begin{array}{cc} {}&{\color{green}x} & {2}& {\color{green}{1}}& {0.5}& {\color{green}{0.1}}& {0.01}& {\color{green}{0.001}}& {0.0001} &{} \\ {}&{\color{green}{f(x)}} & {4}& {\color{green}{1}}& {0.25}& {\color{green}{0.01}}& {0.0001}& {\color{green}{10^{-6}}}& {10^{-8}} &{} \\ \end{array}$

• It is clear that, as x become closer and closer to 0 from the right, f(x) becomes closer and closer to 0
• We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 0$$

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• 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  

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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

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• 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 

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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:
$$\lim_{x\rightarrow 1^{+}} f(x) = 3$$

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• 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

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In the next section, we will see a few more examples. 

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