Limits
In order to really understand Calculus, we first must understand a limit of a function at a point. The notion of a limit is the value the function approaches (y) as we approach a particular value of x. In order for a function to have a limit at a point the value of the function must only approach the same value from the left and the right, not necessarily the value at the point.

More formally: lim x->A+ f(x) = lim x->A- f(x)

We will look at limits from a graph, a table and algebraically with a piecewise function.
Limits from a Graph

Example 1

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Limit at A
Since, lim x -> A+ = 2 and lim x -> A- = 2. OR more simply lim x -> A+ = lim x -> A- , then lim x -> A = 2. The fact that the value of the function at A = 5 has no bearing upon the limit. The limit is what the function APPROACHES, not what the function value is at x = A.

Limit at B
Since, lim x -> B+ = 2 and lim x -> B- = 2, since lim x -> B+ = lim x -> B- , then lim x -> B = 2. 

Example 2

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Limit at A
Since, lim x -> A+ = 3 and lim x -> A- = 1. Since, lim x -> A+ = 3 ≠ lim x -> A- , then no limit exists a x = A.

Limits at Q and R

Point Q only has a limit from the right which is 1, no limit from the left exists since the function does not exist to the left of Q.  In other words, point Q only has a right-handed limit which is 1.  

Similarly, point R only has a limit from the left which is 3, no limit from the right exists since the function does not exist to the right of R.  In other words, point R only has a left-handed limit which is 3.

Example 3

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Limit at 3

Since, lim x -> 3+ = 0 and lim x -> 3- = 0. Since, lim x -> 3+ =  lim x -> 3- , then limit x -> 3 = 0

Limits from a Table
Limits Algebraically for a Piecewise Function I
Limits Algebraically for a Piecewise Function II
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