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| Limits (Contd....) |
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| Limits at infinity |
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If x is a variable such that it can take any real value how much ever  |
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If x is a variable such that it can take any real value how much ever  |
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| Example: |
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 We write |
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| Infinite limits |
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| Let f(x) be a function of x, if the value of f(x) can be made greater than any pre-assigned number by taking x close to 'a', then we say |
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Similarly, if the value of f (x) can be made less than any pre-assigned number by taking x close to 'a', then we say f (x) tends to - as x approaches 'a' |
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| Example: |
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| In the first quadrant, as x approaches zero from the right of zero, the value of f(x) increases without bound. It becomes greater than any assigned positive real value. |
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| In this case, we say |
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| or |
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Similarly, in the third quadrant as x approaches zero from the left, the
value of  decreases without bound. It can be smaller than any pre-assigned negative real number. |
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| The following statement is useful to evaluate limits at infinity of rational functions. |
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| Example: |
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| Evaluate the limit: |
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| Suggested answer: |
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| We have discussed earlier about right hand limit and left hand limit. Both these limits are called one sided limits. |
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| x approaches a from the right side and through values greater than a. |
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| For a function f(x), we say |
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 as left hand Limit, as x approaches
a from the left and through the values lesser than a. |
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| The two important properties of these one-sided limits that |
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| i) If the left hand limit and right hand limit of a function at a point exists, but are not equal, then we conclude that the limit at that point does not exist. |
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| ii) If LHL and RHL of a function at a point (say a) exist and they are equal, we conclude that limit at that point exists and we write |
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| We conclude our discussion on limits with one example on one sided limits. |
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| Example: |
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| For the function f(x) = x+ (x-[x])2, find the RHL and LHL at x = 2. Check whether |
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| Suggested answer: |
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RHL =  |
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| = 2 |
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| = 3 |
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