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 Limits One-Sided Limits The function  f  has the right-hand limit L as x approaches a from the right, written if the values of f(x) can be made as close to L as we please by taking x sufficiently close to (but not equal to) a and to the right of a. Similarly, the function  f  has the left-hand limit M as x approaches a from the left, written if the values of f(x) can be made as close to M as we please by taking x sufficiently close to (but not equal to) a and to the left of a. The connection between one-sided limits and the two-sided limit is given below: Let  f  be a function that is defined for all values of x close to x=a with the possible exception of a itself. Then if and only if If determine whether exists. Solution: Evaluate and then compare the one-sided limits. When , x<2 so f(x)=x2+2; When , x>2 so f(x)=-4x+16; Because ,   does not exist. It is easy to tell from the graph of the function, shown below, that the one-sided limits exists but are not equal. If determine whether exists. Solution: Evaluate and then compare the one-sided limits. When , x>1 so f(x)=x2+1; When , x<1 so f(x)=3x-1; Because ,   .