Left and Right Hand Limits Left and Right Hand Limits While defining the limit of a function f as x tends to 'a', we consider values of f (x) when x is very closed to 'a'. The values of x may be greater or lesser than a. If we restrict x to values less than a, we have seen that it is x + a- or x ® (a - 0). The limit of f with this restriction on x, is called the 'Left Hand Limit'. Similarly if x takes only values greater than a, as seen in x ® a+ or x ® a + 0. The limit of f is then called the 'right hand limit'. Definition :A function 'f' is said to tend to limit l as x tends to a from the left, if for each e > 0 (however small), there exists d > 0 such that | f(x) - l | > e when a - d < x < a In symbols, we then write f (x) = l or
f (x) = l
or f (a - 0) = l or f (a-) = l Know More About: Anti derivative of cosx
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One-Sided Limits In the final two examples in the previous section we saw two limits that did not exist. Given a function f(x) and objective a, the left-hand limit of f(x) as x approaches a has the value L if the values of f(x) get closer and closer to L as for values of x which are to the left of a but increasingly near to a. The right-hand limit of f(x) at a is L if the values of f(x) get closer and closer to L as for values of x which are to the right of a but increasingly near to a. This notation says that the left-hand limit of u/|u| as x approaches 0 is -1 and that the right-hand limit of u/|u| as x approaches 0 is 1. (That fixes Sinis' sinistister scion Smith!) For sqrt(x - 2), the left-hand limit at 2 still does not exist since the function is not defined to the left of 2, but the right-hand limit exists and is equal to 0: Left-handed and right-handed limits are called one-sided limits. Below are two functions h(t) and j(t), fresh out of Smith's Chamber of Cybernetic Cruelty. You can explore the values of h(t) as before, and the graph of j(t) is given below. Your task is to determine the left- and right-hand limits of h(t) and j(t) at the objective t = -4. When you are done, go to the next page. It starts with an equipment check asking you for these one-sided limits. DON'T FORGET TO MAKE A NOTE OF THE LIMITS FIRST. If you need to "try again", you can come back here to gather some more evidence.
Read More About: Antiderivative inverse tangent
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Example:Show that 4x = 8 Solution : For | 4x - 8 | < e i.e. if
if
| 4 (x-2) | < e
4 | x - 2 | < e i.e.,
if
| x - 2 | < e/4
Thus d = e/4 ; we find, therefore, that every e> 0, a number d > 0 where d = e/4, satisfying | 4x - 8 | < e for all | x - 2 | < d Hence 4x = 8
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Thank You
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