In the previous section, we completed a discussion on three dimensional geometry. In this chapter, we will see limits and derivatives.
The basics about limits can be discussed using some examples.
Example 1:
This can be written in 7 steps:
1. Fig.13.1 below shows the graph of the function y = f(x) = x+3
 |
| Fig.13.1 |
• Consider the point (4,0) on the x-axis.
♦ Choose any convenient point (on the x-axis) to the left of (4,0)
• From that convenient point, move (along the x-axis) gradually towards (4,0)
2. As we move, the x values will be increasing: 2, 2.5, 3, 3.5 so on . . .
• The corresponding y values will be:
♦ 2+3 = 5
♦ 2.5+3 = 5.5
♦ 3+3 = 6
♦ 3.5+3 = 6.5
♦ so on . . .
3. We are moving towards (4,0).
• Imagine that, we do not go up to exact (4,0). Instead, we go only to those points which are close to (4,0)
• That means, we go to points like (3.9,0), (3.99,0), (3.999,0) etc.,
• Remember that:
♦ (3.99,0) is closer to (4,0) than (3.9,0)
♦ (3.999,0) is closer to (4,0) than (3.99,0)
✰ We can write: 3.9 < 3.99 < 3.999 < 4
4. The following table 13.1 gives the corresponding y values:
Table 13.1
$\begin{array}{ll}
{}&{x}
& {2}& {2.5}& {3}& {3.5}& {3.9}& {3.99}& {3.999}
&{} \\
{}&{y}
& {5}& {5.5}& {6}& {6.5}& {6.9}& {6.99}& {6.999}
&{} \\
\end{array}$
•
We see that, as we get closer and closer to (4,0), the y values get closer and closer to 7.
•
Remember that:
♦ 6.99 is closer to 7 than 6.9
♦ 6.999 is closer to 7 than 6.99
✰ We can write: 6.9 < 6.99 < 6.999 < 7
5. Let us write an important fact here:
We move closer and closer to a point. But we never reach that exact point.
6. This fact can be demonstrated in two steps:
(i) We went up to (3.999,0). There are three ‘9’ s after the decimal point.
•
We can go to a point where there are a million ‘9’ s after the decimal point. Even then, we cannot say that we have reached (4,0)
•
Having a million '9' s is only very close to (4,0). It is not exact (4,0)
(ii) We obtained y value up to 6.999. There are three ‘9’ s after the decimal point.
•
We can obtain a point where there are a million ‘9’ s after the decimal
point. Even then, we cannot say that we have obtained 7.
•
Having a million '9' s is only very close to 7. It is not exact 7.
7. So from table 13.1, it is clear that:
As we get closer and closer to (4,0), the y value get closer and closer to 7.
•
Mathematically, we write this as:
$$\lim_{x\rightarrow 4^{-}} f(x) = 7$$
•
This is read as:
Limit of f(x) as x approaches 4 from the left equals 7.
(The -ve sign given as superscript above '4' indicates that, x approaches 4 from the left)
We can obtain a similar result by moving towards (4,0) from the right. It can be written in 7 steps:
1. In the Fig.13.1 above, choose any convenient point (on the x-axis) to the right of (4,0)
•
From that convenient point, move (along the x-axis) gradually towards (4,0)
2. As we move, the x values will be decreasing: 6, 5.5, 5, 4.5 so on . . .
•
The corresponding y values will be:
♦ 6+3 = 9
♦ 5.5+3 = 8.5
♦ 5+3 = 8
♦ 4.5+3 = 7.5
♦ so on . . .
3. We are moving towards (4,0).
•
Imagine that, we do not go up to exact (4,0). Instead, we go only to those points which are close to (4,0)
•
That means, we go to points like (4.2,0), (4.1,0), (4.01,0), (4.001,0), (4.0001,0) etc.,
•
Remember that:
♦ (4.0001,0) is closer to (4,0) than (4.001,0)
♦ (4.001,0) is closer to (4,0) than (4.01,0)
✰ We can write: 4 < 4.0001 < 4.001 < 4.01 < 4.1
4. The following table 13.2 gives the corresponding y values:
Table 13.2
$\begin{array}{ll}
{}&{x}
& {6}& {5.5}& {5}& {4.5}& {4.1}& {4.01}& {4.001}
&{4.0001} \\
{}&{y}
& {9}& {8.5}& {8}& {7.5}& {7.1}& {7.01}& {7.001}
&{7.0001} \\
\end{array}$
•
We see that, as we get closer and closer to (4,0), the y values get closer and closer to 7.
•
Remember that:
♦ 7.0001 is closer to 7 than 7.001
♦ 7.001 is closer to 7 than 7.01
✰ We can write: 7 < 7.0001 < 7.001 < 7.01 < 7.1
5. Let us write an important fact here:
We move closer and closer to a point. But we never reach that exact point.
6. This fact can be demonstrated in two steps:
(i) We went (from right to left) up to (4.0001,0). There are three ‘zeros’ after the decimal point.
•
We can go to a point where there are a million ‘zeros’ after the decimal
point. Even then, we cannot say that we have reached (4,0)
•
Having a million 'zeros' is only very close to (4,0). It is not exact (4,0)
(ii) We obtained y value up to 7.0001. There are three ‘zeros’ after the decimal point.
•
We can obtain a point where there are a million ‘zeros’ after the decimal
point. Even then, we cannot say that we have obtained 7.
•
Having a million 'zeros' is only very close to 7. It is not exact 7.
7. So from table 13.2, it is clear that:
As we get closer and closer to (4,0), the y value get closer and closer to 7.
•
Mathematically, we write this as:
$$\lim_{x\rightarrow 4^{+}} f(x) = 7$$
•
This is read as:
Limit of f(x) as x approaches 4 from the right equals 7.
(The +ve sign given as superscript above '4' indicates that, x approaches 4 from the right)
• In the above example, we see that:
♦ Left side limit of f(x) at 4 is 7
♦ Right side limit of f(x) at 4 is 7
♦ Value of f(x) at 4 is also 7
✰ That means, limit at 4 can be obtained by putting x = 4.
✰ f(x) = x+3
✰ f(4) = 4 + 3 = 7
Example 2:
In this example, we use the same function that we saw in example 1. It can be written in 5 steps:
1. Fig.13.2 below shows the graph of the function y = f(x) = x + 3
 |
| Fig.13.2 |
•
Point B(6,9) is marked with an ordinary circle.
•
We know that:
♦ Points marked with filled circle are included in the calculations
♦ Points marked with ordinary circle are not included in the calculations.
•
For example, the interval 2 ≤ n < 5 can be marked on a number line, with a filled circle at 2 and an ordinary circle at 5.
2. In our present case, B(6,9) is to be avoided while plotting the function f(x) = x + 3
•
We can say that, the function is not defined at x = 6
3. Let us find the limit of f(x) as x approaches 6 from the left. For that, we prepare table 13.3 below:
Table 13.3
$\begin{array}{ll}
{}&{x}
& {4.5}& {5}& {5.5}& {5.8}& {5.9}& {5.99}& {5.999}
&{} \\
{}&{y}
& {7.5}& {8}& {8.5}& {8.8}& {8.9}& {8.99}& {8.999}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 6 from the left, f(x) becomes closer and closer to 9
•
We can write:
$$\lim_{x\rightarrow 4^{-}} f(x) = 9$$
4. Let us find the limit of f(x) as x approaches 6 from the right. For that, we prepare table 13.4 below:
Table 13.4
$\begin{array}{ll}
{}&{x}
& {8}& {7.5}& {7}& {6.5}& {6.1}& {6.01}& {6.001}
&{} \\
{}&{y}
& {11}& {10.5}& {10}& {9.5}& {9.1}& {9.01}& {9.001}
&{} \\
\end{array}$
•
It is clear that, as x become closer and closer to 6 from the right, f(x) becomes closer and closer to 9
•
We can write:
$$\lim_{x\rightarrow 4^{+}} f(x) = 9$$
5. We note an important point here. It can be written in steps:
(i) Based on the graph in fig.13.2, f(x) can never have the value ‘9’
(ii) But the limiting values like 8.999, 8.9999, 9.0001, 9.00001 etc., are allowed.
(iii) When x approaches 6 from either sides, f(x) approaches 9
So the limit when x approaches 6 is indeed 9
(iv) We can write:
♦ f(x) = 9 is not allowed.
♦ $\lim_{x\rightarrow 6^{-}} f(x) = 9$ is allowed.
♦ $\lim_{x\rightarrow 6^{+}} f(x) = 9$ is allowed.
• In this example, we see that:
♦ Left side limit of f(x) at 6 is 9
♦ Right side limit of f(x) at 6 is 9
♦ Though not allowed, the value of f(x) at 6 is also 9
✰ That means, limit at 6 can be obtained by putting x = 6.
✰ f(x) = x+3
✰ f(6) = 6 + 3 = 9
In the next section, we will see a few more examples.
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