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 One-Sided Limits The function  f  has the right-hand limit L as x approaches a from the right, written $\lim_{x\to\text{a^+}}\text{f(x)}=L$ 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 $\lim_{x\to\text{a^-}}\text{f(x)}=M$ 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 $\lim_{x\to\text{a}}\text{f(x)}=L$ if and only if $\lim_{x\to\text{a^-}}\text{f(x)}=\lim_{x\to\text{a^+}}\text{f(x)}=L$ If determine whether $\lim_{x\to\text{2}}\text{f(x)}$ exists. Solution: Evaluate and then compare the one-sided limits. When $x\to\text{2^-}$, x<2 so f(x)=x2+2; $\lim_{x\to\text{2^-}}\text{f(x)}=\lim_{x\to\text{2}}\text{x^2+2}=2^2+2=6$ When $x\to\text{2^+}$, x>2 so f(x)=-4x+16; $\lim_{x\to\text{2^+}}\text{f(x)}=\lim_{x\to\text{2}}\text{-4x+16}=-4(2)+16=8$ Because $\lim_{x\to\text{2^-}}\text{f(x)}\;\neq\;\lim_{x\to\text{2^+}}$,   $\lim_{x\to\text{2}}\text{f(x)}$ 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 $\lim_{x\to\text{1}}\text{f(x)}$ exists. Solution: Evaluate and then compare the one-sided limits. When $x\to\text{1^+}$, x>1 so f(x)=x2+1; $\lim_{x\to\text{1^+}}\text{f(x)}=\lim_{x\to\text{1}}\text{x^2+1}=1^2+1=2$ When $x\to\text{1^-}$, x<1 so f(x)=3x-1; $\lim_{x\to\text{1^-}}\text{f(x)}=\lim_{x\to\text{1}}\text{3x-1}=3(1)-1=2$ Because $\lim_{x\to\text{1^-}}\text{f(x)}\;=\;\lim_{x\to\text{1^+}}=2$,   $\lim_{x\to\text{1}}\text{f(x)}=2$.