Defining the Limiting Process

Limits play an important role in probing the behaviour of a function at any point in its domain, and the notation we use to describe this process is  

in this symbolism, the suffix to the symbol lim indicates that, although x approaches a, it never actually takes the value a. For the limit to exist, the same (finite) result must be obtained whether we approach a from smaller or larger values of x. Furthermore, if m=f a), then the function is said to be continuous at x = a. 

Taking even smaller increments either side of 3, say x = 3 + 0.0001, we find that /(3.0001) = 6.0001 and /(2.9999) = 5.9999. These results suggest that for smaller and smaller increments in x, either side of x = 3, the values of the function become closer and closer to 6. Thus we say that, in the limit as x — 3, m takes the value 6  

In practice, it is often easiest when evaluating limits to write x = a + 5, and consider what happens as 6 — 0, but never takes the value zero. This procedure allows us to let x become as close as we like to the value a, without it taking the value x = a. 

For each of the following functions, f(x), identify any points of discontinuity (those values of x where the function is of indeterminate value) and use the method described in Worked Problem 3.1, where appropriate, to find the limiting values of the following functions at your suggested points of discontinuity. 

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