In the previous section, we completed a discussion on the examples which demonstrated the basics of limits. In this section, we will see algebra of limits.
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In the previous sections, we have seen some examples in which:
♦ sum of the limits of two functions
♦ is equal to
♦ the limit of the sum of the two functions.
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For example:
$$\lim_{x\rightarrow 1} x^2~+~\lim_{x\rightarrow 1} x~=~\lim_{x\rightarrow 1} \left[x^2 + x \right]$$
• In the second chapter, we saw that:
♦ Two functions can be added together.
♦ One function can be subtracted from another function.
♦ A function can be multiplied by another function.
♦ A function can be divided by another function.
(Those details can be seen here)
◼ Now we will write a theorem of the limiting process related to addition, subtraction, multiplication and division of functions.
Theorem 1
This can be written in 4 steps:
Let f and g be two functions such that, both $\lim_{x\rightarrow a} f(x)$ and $\lim_{x\rightarrow a} g(x)$ exist.
Then:
1. Addition of individual limits is possible:
♦ Limit of the sum [f(x) + g(x)]
♦ is equal to
♦ the sum of the two individual limits.
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That is:
$$\lim_{x\rightarrow a} [f(x) + g(x)]=\lim_{x\rightarrow a} f(x)+\lim_{x\rightarrow a} g(x)$$
2. Difference of individual limits is possible:
♦ Limit of the difference [f(x) - g(x)]
♦ is equal to
♦ the difference of the two individual limits.
•
That is:
$$\lim_{x\rightarrow a} [f(x) - g(x)]=\lim_{x\rightarrow a} f(x)-\lim_{x\rightarrow a} g(x)$$
3. Multiplication of individual limits is possible:
♦ Limit of the product [f(x) . g(x)]
♦ is equal to
♦ the product of the two individual limits.
•
That is:
$$\lim_{x\rightarrow a} [f(x) \, . \, g(x)]=\lim_{x\rightarrow a} f(x) \, .\, \lim_{x\rightarrow a} g(x)$$
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Here there is special case also. It can be written in 2 steps:
(i) One of the function, say g(x) is a constant function $\lambda$.
♦ That is., $g(x) = \lambda$
♦ Where $\lambda$ is a real number.
(ii) Then we get:
$$\lim_{x\rightarrow a} [\lambda \, . \, f(x)]=\lambda \, .\, \lim_{x\rightarrow a} f(x)$$
4. Division of individual limits is possible:
♦ Limit of the quotient $\frac{f(x)}{g(x)}$
♦ is equal to
♦ the quotient of the two individual limits.
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That is:
$$\lim_{x\rightarrow a} \left[\frac{f(x)}{g(x)} \right] = \frac{\lim_{x\rightarrow a} f(x)}{\lim_{x\rightarrow a} g(x)}$$
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In this fraction, the denominator should not be equal to zero.
We will see the proof for the above theorem in higher classes. At present, we will see some examples.
Example for addition:
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This can be written in 5 steps:
1. Let us write the functions:
(i) Let the first function be: f(x) = x3 + 1
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This is plotted in green color in fig.13.13 below.
(ii) Let the second function be: g(x) = x2 + 2x + 3
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This is plotted in yellow color in fig.13.13 below.
(iii) So the sum of the two functions will be:
f(x) + g(x) = x3 + x2 + 2x + 4
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This is plotted in red color in fig.13.13 below.
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| Fig.12.13 |
2. Let us write the limit of f(x) as x approaches 1
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We get: $\lim_{x\rightarrow 1} f(x) = (1^3 + 1) = (1+1) = 2$
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Indeed, we have a green filled circle at (1,2)
3. Let us write the limit of g(x) as x approaches 1
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We get: $\lim_{x\rightarrow 1} g(x) = (1^2 + 2 × 1 + 3) = (1+2+3) = 6$
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Indeed, we have a yellow filled circle at (1,6)
4. Let us write the limit of [f(x) + g(x)] as x approaches 1
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We get: $\lim_{x\rightarrow 1} [f(x) + g(x)] = (1^3 + 1^2 + 2 × 1 + 4) = (1+1+2+4) = 8$
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Indeed, we have a red filled circle at (1,8)
5. Comparing the above steps (2), (3) and (4), we can write:
$$\lim_{x\rightarrow 1} [f(x) + g(x)]= 8 = \lim_{x\rightarrow 1} f(x)+\lim_{x\rightarrow 1} g(x) = 2 + 6$$
Example for subtraction:
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This can be written in 5 steps:
1. Let us write the functions:
(i) Let the first function be: f(x) = x3 + 1
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This is plotted in green color in fig.13.14 below.
(ii) Let the second function be: g(x) = x2 + 2x + 3
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This is plotted in yellow color in fig.13.14 below.
(iii) So the difference of the two functions will be:
f(x) - g(x) = x3 - x2 - 2x - 2
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This is plotted in red color in fig.13.14 below.
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| Fig.13.14 |
2. Let us write the limit of f(x) as x approaches 1
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We get: $\lim_{x\rightarrow 1} f(x) = (1^3 + 1) = (1+1) = 2$
•
Indeed, we have a green filled circle at (1,2)
3. Let us write the limit of g(x) as x approaches 1
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We get: $\lim_{x\rightarrow 1} g(x) = (1^2 + 2 × 1 + 3) = (1+2+3) = 6$
•
Indeed, we have a yellow filled circle at (1,6)
4. Let us write the limit of [f(x) - g(x)] as x approaches 1
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We get: $\lim_{x\rightarrow 1} [f(x) - g(x)] = (1^3 - 1^2 - 2 × 1 - 2) = (1-1-2-2) = -4$
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Indeed, we have a red filled circle at (1,-4)
5. Comparing the above steps (2), (3) and (4), we can write:
$$\lim_{x\rightarrow 1} [f(x) - g(x)]= -4 = \lim_{x\rightarrow 1} f(x)-\lim_{x\rightarrow 1} g(x) = 2 - 6$$
In the next section, we will see examples for multiplication and division.
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