Definition of a limit: If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.
lim f(x) = L
x--> c
lim (3x- 2)
x--> 2Let f(x) = 3x- 2. Then construct a table that shows values of f(x) when x is close to 2.
x f(x)
1.9 3.7
1.99 3.97
x--> 1+ (the + meaning that x approaches 1 from the right; right-hand limit)
x--> 1- (the - meaning that x approaches 1 from the left; left-hand limit)
Limits That Fail to Exist:
1.999 3.997
2.0 ?
2.001 4.003
2.01 4.03
2.1 4.3
From the table, you can see that the closer x gets to 2, the closer f(x) gets to 4. So you can estimate the limit to be 4. For this equation you can substitute 2 for x to obtain the limit, so:
lim (3x- 2) = 3(2) -2 = 4
x--> 2
x--> 1+ (the + meaning that x approaches 1 from the right; right-hand limit)
x--> 1- (the - meaning that x approaches 1 from the left; left-hand limit)
Limits That Fail to Exist:
1.) When the one-sided limits are not equal
ex.
2.) Unbounded Behavior
ex.
When x > 100 or x<-100
3.) Oscillating Behavior
ex.
WARNING: Use the above information with caution.
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