In the previous section, we saw the theorem related to algebra of limits. We also saw examples of addition and subtraction of individual limits. In this section, we will see examples for multiplication and division.
Example for multiplication:
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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) = x + 4
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This is plotted in green color in fig.13.15 below.
(ii) Let the second function be: g(x) = x2 + 1
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This is plotted in yellow color in fig.13.15 below.
(iii) So the product of the two functions will be:
f(x) . g(x) = (x+4)(x2+1) = (x3 + x + 4x2 + 4) = x3 + 4x2 + x + 4
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This is plotted in red color in fig.13.15 below.
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| Fig.13.15 |
2. Let us write the limit of f(x) as x approaches -2
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We get: $\lim_{x\rightarrow -2} f(x) = (-2 + 4) = 2$
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Indeed, we have a green filled circle at (-2,2)
3. Let us write the limit of g(x) as x approaches -2
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We get: $\lim_{x\rightarrow -2} g(x) = [(-2)^2 + 1] = (4+1) = 5$
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Indeed, we have a yellow filled circle at (-2,5)
4. Let us write the limit of [f(x) . g(x)] as x approaches -2
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We get: $\lim_{x\rightarrow -2} [f(x) . g(x)] = [(-2)^3 + 4(-2)^2 -2 + 4] = (-8+16-2+4) = 10$
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Indeed, we have a red filled circle at (-2,10)
5. Comparing the above steps (2), (3) and (4), we can write:
$$\lim_{x\rightarrow -2} [f(x) . g(x)]= 10 = \lim_{x\rightarrow -2} f(x) \, . \, \lim_{x\rightarrow -2} g(x) = 2 \, . \, 5$$
Example for the special case of multiplication:
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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) = x2 + 1
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This is plotted in green color in fig.13.16 below.
(ii) Let the second function be: g(x) = 3
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This is plotted in yellow color in fig.13.16 below.
(iii) So the product of the two functions will be:
f(x) . g(x) = (x2+1).3 = 3x2 + 3
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This is plotted in red color in fig.13.16 below.
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| Fig.13.16 |
2. Let us write the limit of f(x) as x approaches 1
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We get: $\lim_{x\rightarrow 1} f(x) = (1^2 + 1) = (1+1) =2$
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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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Here g(x) is a constant function. The constant $\lambda$ = 3.
♦ So whatever be the value of x, the limiting value will be '3'.
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We get: $\lim_{x\rightarrow 1} g(x) = 3$
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The point (1,3) is marked with a yellow filled circle.
4. Let us write the limit of [f(x) . g(x)] as x approaches -2
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We get: $\lim_{x\rightarrow 1} [f(x) . g(x)] = [3(1)^2 + 3] = (3+3) = 6$
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Indeed, we have a red filled circle at (1,6)
5. Comparing the above steps (2), (3) and (4), we can write:
$$\lim_{x\rightarrow 1} [f(x) . g(x)]= 6 = \lim_{x\rightarrow 1} f(x) \, . \, \lim_{x\rightarrow 1} g(x) = 2 \, . \, 3 = \lambda \, . \, \lim_{x\rightarrow 1} f(x) = 3 \, . \, 2$$
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That means:
♦ We need not multiply the two functions.
♦ We just need to take the limit of f(x) and then multiply it by $\lambda$
Example for division:
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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) = x2 + 1
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This is plotted in green color in fig.13.17 below.
(ii) Let the second function be: g(x) = x + 4
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This is plotted in yellow color in fig.13.17 below.
(iii) So the quotient of the two functions will be:
$\frac{f(x)}{g(x)}~=~\frac{x^2 + 1}{x+4}$
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This is plotted in red color in fig.13.17 below.
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| Fig.13.17 |
2. Let us write the limit of f(x) as x approaches -2
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We get: $\lim_{x\rightarrow -2} g(x) = [(-2)^2 + 1] = (4+1) = 5$
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Indeed, we have a green filled circle at (-2,5)
3. Let us write the limit of g(x) as x approaches -2
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We get: $\lim_{x\rightarrow -2} f(x) = (-2 + 4) = 2$
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Indeed, we have a yellow filled circle at (-2,2)
4. Let us write the limit of $\frac{f(x)}{g(x)}$ as x approaches -2
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We get: $\lim_{x\rightarrow -2}\left [\frac{f(x)}{g(x)}\right] = \frac{(-2)^2 +1}{-2+4} = \frac{5}{2} = 2.5$
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Indeed, we have a red filled circle at (-2,2.5)
5. Comparing the above steps (2), (3) and (4), we can write:
$$\lim_{x\rightarrow -2} \left [\frac{f(x)}{g(x)}\right] = 2.5 = \frac{\lim_{x\rightarrow -2} f(x)}{\lim_{x\rightarrow -2} g(x)} = \frac{5}{2}$$
• We have completed the examples for all the four cases.
♦ We have taken the limits at only a few x-values like -2 and 1.
♦ The reader may check other convenient x-values like -5, -0.5, 3.75 etc.,
• In the next section, we will see limits of polynomials and rational functions.
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