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3.3 One-Sided Limits
1. If x approaches the value of a from the left (written as 𝑥𝑥 → 𝑎𝑎 ) and if the values of 𝑓𝑓 approach a real value, L, 𝑓𝑓 ( 𝑥𝑥) → 𝐿𝐿 , then this value L is called a one-sided limit or the hand limit (LHL) of 𝑓𝑓 𝑥𝑥 . In notation form, 𝑥𝑥→𝑎𝑎 lim 𝑓𝑓 ( 𝑥𝑥) = 𝐿𝐿 a 𝑥𝑥 → 𝑎𝑎 𝑥𝑥→𝑎𝑎 + 2. If x approaches the value of a from the right (written as 𝑥𝑥 → 𝑎𝑎 + ) and if the values of 𝑓𝑓 approach a real value, M, 𝑓𝑓 ( 𝑥𝑥) → 𝑀𝑀 , then this value M is called a one-sided limit or the hand limit (RHL) of 𝑓𝑓 ( 𝑥𝑥 ) . In notation form, lim 𝑓𝑓 ( 𝑥𝑥) = a 𝑥𝑥 → 𝑎𝑎 + ( 𝑥𝑥 ) left- ( 𝑥𝑥) right- 𝑀𝑀 2
x a lim f ( x ) = L As the x values get closer to a from values to the left of a On the given function f(x) The y values get closer to L = x a + As the x values get closer to a from values to the right of a On the given function f(x) lim f ( x ) M The y values get closer to M 3
Suppose f(x) is given by the graph below. a) Find lim f ( x ) b) Find x 1 lim f ( x ) 1 + 4 x
a) lim f ( x ) = 1 5 x 1
b) 6 lim f ( x ) = 2 x 1 +
3. Existence of Limits 7 (a) both one-sided limits exist; Therefore limit exists because the left-hand and right-hand limits have the same values. Hence 4. All the properties of (ordinary) limits can also be applied for one sided limits. 𝑥𝑥→ lim 𝑓𝑓 ( 𝑥𝑥) exists if and only if; 𝑥𝑥→𝑎𝑎 (i) LHL, lim 𝑓𝑓 ( 𝑥𝑥) = 𝐿𝐿 (ii) RHL, 𝑥𝑥→𝑎𝑎 + lim 𝑓𝑓 ( 𝑥𝑥) = 𝑀𝑀 𝑥𝑥→𝑎𝑎 (b) both one-sided limits equal (says L=M ), that is if lim 𝑓𝑓 ( 𝑥𝑥) = lim 𝑓𝑓 ( 𝑥𝑥 ) 𝑥𝑥→𝑎𝑎 + 𝐿𝐿 = 𝑀𝑀 lim 𝑓𝑓 ( 𝑥𝑥) = 𝐿𝐿 𝑥𝑥→𝑎𝑎
lim f ( x ) lim f ( x ) x 1 x 1 + 8 𝑥𝑥→7 𝑥𝑥→7 + lim 𝑓𝑓 ( 𝑥𝑥 ) = lim 𝑓𝑓 ( 𝑥𝑥) = 8 Therefore lim 𝑓𝑓 ( 𝑥𝑥 ) exists .

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