In the previous section, and the section before that, we saw some examples which demonstrated the basics of limits. In this section, we will see a few more examples.
Example 8:
This can be written in 3 steps:
1. Fig.13.8 below shows the graph of the function y = f(x) = x3
 |
| Fig.13.8 |
2. Let us find the limit of f(x) as x approaches 1 from the left. For that, we prepare table 13.15 below:
Table 13.15
$\begin{array}{cc}
{}&{\color{green}x}
& {-1}&
{\color{green}{-0.5}}& {0}& {\color{green}{0.5}}&
{0.9}& {\color{green}{0.99}}& {0.999}
&{} \\
{}&{\color{green}{f(x)}}
&
{-1}& {\color{green}{-0.125}}& {0}& {\color{green}{0.125}}&
{0.729}& {\color{green}{0.970}}& {0.997}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the left, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow 1^{-}} f(x) = 1$$
3. Let us find the limit of f(x) as x approaches 1 from the right. For that, we prepare table 13.16 below:
Table 13.16
{}&{\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)}}
& {27}& {\color{green}{8}}& {3.375}& {\color{green}{1.331}}& {1.030301}& {\color{green}{1.003003001}}& {1.000300030001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the right, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow 1^{+}} f(x) = 1$$
• In this example, we see that:
♦ Left side limit of f(x) at 1 is 1
♦ Right side limit of f(x) at 1 is 1
♦ Value of f(x) at 1 is also 1
✰ That means, limit at 0 can be obtained by putting x = 1.
✰ f(x) = x3
✰ f(1) = 13 = 1
Example 9
This can be written in 3 steps:
1. Fig.13.9 below shows the graph of the function: $f(x)~=~\frac{1}{x^2}~\text{for}~x>0$
 |
| Fig.13.9 |
2. Suppose that, we want to find the limit of f(x) at x = 0
• We usually prepare the first table in which x approaches 0 from the left.
• But in this problem, it is given that, 'x > 0'.
♦ That means, f(x) is to be evaluated for all x which are greater than zero.
♦ That means, f(x) is to be evaluated for all x which are +ve real numbers.
• So we need not prepare the table for the left side because, all x values to the left of zero are -ve real numbers.
3. Let us find the limit of f(x) as x approaches 0 from the right. For that, we prepare table 13.17 below:
Table 13.17
$\begin{array}{cc}
{}&{\color{green}x}
& {2}&
{\color{green}{1}}& {0.5}& {\color{green}{0.1}}&
{0.01}& {\color{green}{0.001}}& {0.0001}
&{} \\
{}&{\color{green}{f(x)}}
&
{0.25}& {\color{green}{1}}& {4}& {\color{green}{100}}&
{10000}& {\color{green}{10^6}}& {10^8}
&{} \\
\end{array}$
•
It is clear that:
♦ x become closer and closer to 0 from the right.
♦ f(x) becomes larger and larger. f(x) becomes infinitely large.
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) =+ \infty$$
Example 10
This can be written in 3 steps:
1. Fig.13.10 below shows the graph of the function: f(x) = x2 + x
 |
| Fig.13.10 |
2. Let us find the limit of f(x) as x approaches 1 from the left. For that, we prepare table 13.18 below:
Table 13.18
$\begin{array}{cc}
{}&{\color{green}x}
& {-1}&
{\color{green}{-0.5}}& {0}& {\color{green}{0.5}}&
{0.9}& {\color{green}{0.99}}& {0.999}
&{} \\
{}&{\color{green}{f(x)}}
&
{0}& {\color{green}{-0.25}}& {0}& {\color{green}{0.75}}&
{1.71}& {\color{green}{1.9701}}& {1.997001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the left, f(x) becomes closer and closer to 2
•
We can write:
$$\lim_{x\rightarrow 1^{-}} f(x) = 2$$
3. Let us find the limit of f(x) as x approaches 1 from the right. For that, we prepare table 13.19 below:
Table 13.19
{}&{\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)}}
& {12}& {\color{green}{6}}& {3.75}& {\color{green}{2.31}}& {2.0301}& {\color{green}{2.003001}}& {2.00030001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 1 from the right, f(x) becomes closer and closer to 2
•
We can write:
$$\lim_{x\rightarrow 1^{+}} f(x) = 2$$
• In this example, we see that:
♦ Left side limit of f(x) at 1 is 2
♦ Right side limit of f(x) at 1 is 2
♦ Value of f(x) at 1 is also 2
✰ That means, limit at 1 can be obtained by putting x = 1.
✰ f(x) = x2 + x
✰ f(1) = 12 + 1 = 2
Based on this example, we can derive two useful results:
Result I:
This can be written in 5 steps:
1. We have:
$$\lim_{x\rightarrow 1} x^2 = 1$$
• We can obtain this limit from the graph in fig.13.3 of section 13.1.
2. We have:
$$\lim_{x\rightarrow 1} x = 1$$
• We can obtain this limit from the graph of the function y = f(x) = x.
3. Adding the above two limits, we get:
$$\lim_{x\rightarrow 1} x^2~+~\lim_{x\rightarrow 1} x~=~1+1~=~2$$
4. But from example 10 above, we got:
$$\lim_{x\rightarrow 1} \left(x^2 + x \right) = 2$$
5. So we can equate (3) and (4). We get:
$$\lim_{x\rightarrow 1} x^2~+~\lim_{x\rightarrow 1} x~=~\lim_{x\rightarrow 1} \left[x^2 + x \right]$$
Result II:
This can be written in 5 steps:
1. We have:
$$\lim_{x\rightarrow 1} (x+1) = 2$$
2. While deriving result I above, we saw that:
$$\lim_{x\rightarrow 1} x = 1$$
3. Multiplying the above two limits, we get:
$$\lim_{x\rightarrow 1} x . \lim_{x\rightarrow 1} (x+1)~=~1 . 2~=~2$$
4. But from example 10 above, we got:
$$\lim_{x\rightarrow 1} \left(x^2 + x \right) = 2$$
5. So we can equate (3) and (4). We get:
$$\lim_{x\rightarrow 1} x . \lim_{x\rightarrow 1} (x+1)~=~\lim_{x\rightarrow 1} \left[x(x+1) \right]~=~\lim_{x\rightarrow 1} \left[x^2 + x \right]$$
Example 11
This can be written in 3 steps:
1. Fig.13.11 below shows the graph of the function: f(x) = sin x
 |
| Fig.13.11 |
2. Let us find the limit of f(x) as x approaches $\frac{\pi}{2}$ from the left. For that, we prepare table 13.20 below:
Table 13.20
$\begin{array}{cc}
{}&{\color{green}x}
& {\frac{\pi}{2.8}}&
{\color{green}{\frac{\pi}{2.5}}}& {\frac{\pi}{2.2}}& {\color{green}{\frac{\pi}{2.1}}}&
{\frac{\pi}{2.01}}& {}& {}
&{} \\
{}&{\color{green}{f(x)}}
&
{0.90097}& {\color{green}{0.951006}}& {0.98982}& {\color{green}{0.9972}}&
{0.99997}& {}& {}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to $\frac{\pi}{2}$ from the left, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow {\frac{\pi}{2}}^{-}} f(x) = 1$$
3. Let us find the limit of f(x) as x approaches $\frac{\pi}{2}$ from the right. For that, we prepare table 13.21 below:
Table 13.21
$\begin{array}{cc}{}&{\color{green}x}
& {\frac{\pi}{1.5}}& {\color{green}{\frac{\pi}{1.8}}}& {\frac{\pi}{1.9}}& {\color{green}{\frac{\pi}{1.99}}}& {}& {}& {}
&{} \\
{}&{\color{green}{f(x)}}
& {0.8660}& {\color{green}{0.9848}}& {0.9965}& {\color{green}{0.9999}}& {}& {}& {}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to $\frac{\pi}{2}$ from the right, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow {\frac{\pi}{2}}^{+}} f(x) = 1$$
• In this example, we see that:
♦ Left side limit of f(x) at $\frac{\pi}{2}$ is 1
♦ Right side limit of f(x) at $\frac{\pi}{2}$ is 1
♦ Value of f(x) at $\frac{\pi}{2}$ is also 1
✰ That means, limit at $\frac{\pi}{2}$ can be obtained by putting x = $\frac{\pi}{2}$.
✰ f(x) = sin x
✰ $f \left(\frac{\pi}{2} \right)~=~\sin \left(\frac{\pi}{2} \right)~=~1$
Example 12
This can be written in 3 steps:
1. Fig.13.12 below shows the graph of the function: f(x) = x + cos x
 |
| Fig.13.12 |
2. Let us find the limit of f(x) as x approaches 0 from the left. For that, we prepare table 13.22 below:
Table 13.22
$\begin{array}{cc}
{}&{\color{green}x}
& {-\frac{\pi}{1.5}}&
{\color{green}{-\frac{\pi}{2.0}}}& {-\frac{\pi}{5.0}}& {\color{green}{-\frac{\pi}{10}}}&
{-\frac{\pi}{100}}& {\color{green}{-\frac{\pi}{1000}}}& {-\frac{\pi}{10000}}
&{} \\
{}&{\color{green}{f(x)}}
&
{-2.5944}& {\color{green}{-1.5708}}& {0.1807}& {\color{green}{0.6369}}&
{0.9681}& {\color{green}{0.9968}}& {0.9997}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the left, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow 0^{-}} f(x) = 1$$
•
We can note an interesting fact here. It can be written in 3 steps:
(i) From the graph in fig.13.12 above, we see that, f(x) is mostly -ve on the left side of the y-axis. But just before reaching the y-axis, it becomes +ve.
(ii) Indeed, in table 13.22 above, we see that, the last five f(x) values are +ve.
(iii) This fact helps us to appreciate the relation between a graph and the table used to prepare that graph. However, this fact do not have much significance on our present discussion about limits.
3. Let us find the limit of f(x) as x approaches 1 from the right. For that, we prepare table 13.23 below:
Table 13.23
$\begin{array}{cc}
{}&{\color{green}x}
& {\frac{\pi}{1.5}}&
{\color{green}{\frac{\pi}{2.0}}}& {\frac{\pi}{5.0}}& {\color{green}{\frac{\pi}{10}}}&
{\frac{\pi}{100}}& {\color{green}{\frac{\pi}{1000}}}& {\frac{\pi}{10000}}
&{} \\
{}&{\color{green}{f(x)}}
&
{1.5944}& {\color{green}{1.5708}}& {1.4373}& {\color{green}{1.2652}}&
{1.0309}& {\color{green}{1.0031}}& {1.0003}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 0 from the right, f(x) becomes closer and closer to 1
•
We can write:
$$\lim_{x\rightarrow 0^{+}} f(x) = 1$$
• 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 1
♦ Value of f(x) at 0 is also 1
✰ That means, limit at 0 can be obtained by putting x = 0.
✰ f(x) = x + cos x
✰ f(0) = (0 + cos 0) = (0 + 1) = 1
From this example, we can derive a result similar to the result I that we derived earlier in this section. It can be written in 5 steps:
1. We have:
$$\lim_{x\rightarrow 0} x = 0$$
2. We have:
$$\lim_{x\rightarrow 0} cos x = 1$$
• We can obtain this limit from the graph of the function y = f(x) = cos x.
3. Adding the above two limits, we get:
$$\lim_{x\rightarrow 0} x~+~\lim_{x\rightarrow 0} cos x~=~0+1~=~1$$
4. But from this example 12 , we got:
$$\lim_{x\rightarrow 0} \left(x + cos x \right) = 1$$
5. So we can equate (3) and (4). We get:
$$\lim_{x\rightarrow 0} x~+~\lim_{x\rightarrow 0} cos x~=~\lim_{x\rightarrow 0} \left[x + cos x \right]$$
In the next section, we will see algebra of limits.
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