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| Limits |
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| But we are interested in finding the value of y near 2. |
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| When |
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| x = 1.9, y = 3.9 |
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| x = 1.99, y = 3.99 |
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| x = 1.999 y = 3.999 |
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| . . |
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| . . |
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| . . |
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| Similarly, when |
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| x = 2.1 y = 4.1 |
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| x = 2.01 y = 4.01 |
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| x = 2.001 y = 4.001 |
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| . . |
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| . . |
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| Observe that as x approaches 2, y approaches 4. 4 is called the limit of f(x) as x approaches 2 and is represented as |
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 If x is close to 2, f(x) is close to 4. |
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If |x - 2| is small then |f(x) - 4| is also small. |
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| Note 2: When x approaches 2 from left, the limit obtained is called left hand limit and when x approaches 2 from the right, the limit obtained is called the right hand limit. |
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Note 3: We say that  because both left
hand limit and |
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| right hand limit are finite and equal. |
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| If there exists a real number l such that if |f (x) - l| can be made as small as we possible by taking x sufficiently close to a, then l is called the limit of f (x) as x tends to 'a'. |
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| Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit which is symbolically written as |
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| Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit which is symbolically written as |
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| Note: |
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| We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal. |
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| Limit of composite functions |
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| a) Limits of polynomial functions can be found by substitution |
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| If f(x) = anxn + an-1xn-1+an-2xn-2+…….. a0, then |
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| Example: |
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| b) Limit of a rational function can be found by substitution, if the limit of the denominator is not zero. |
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| In other words, |
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| where P(x) and Q(x) are polynomials, then we have |
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| Example: |
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| c) If Q(c)=0, common factors between the numerator and denominator are identified and cancelled. This reduces the rational function to another rational function whose denominator is not zero at c. |
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| Example: |
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| If Q(c) = 0, we can also factorise and cancel a common factor. |
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| Example: |
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| Suggested answer: |
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| d) In the rational function, we have |
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| If P(c) = a, Q(c ) = 0, then |
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| Example: |
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| At x =1, the numerator of the rational function 6, but denominator is 0. |
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| Theorem 1: |
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| Case (i): If n is a positive integer. |
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| We have, |
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| Case (ii): If n is a negative integer. |
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| Let n = -k, where k is a positive integer |
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| Hence true for negative integers. |
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| Case (iii): When n is a rational number. |
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| Hence the theorem. |
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