[tex]f(x)=\begin{cases}5x-8&\text{for }x<0\\|-4-x|&\text{for }x\ge0\end{cases}[/tex]
The limits from either side are
[tex]\displaystyle\lim_{x\to0^-}f(x)=\lim_{x\to0}(5x-8)=-8[/tex]
[tex]\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0}|-4-x|=\lim_{x\to0}|x+4|=|4|=4[/tex]
The one-sided limits don't match, so the limit as [tex]x\to0[/tex] does not exist.
A function assigns the values. The limit as x→0 does not exist.
A function assigns the value of each element of one set to the other specific element of another set.
The given limits can be written as,
[tex]f(x)=\left \{{5x-8\ \ \ {\rm for\ x < 0} \atop |-4-x|\ \ \ {\rm for\ x \geq 0}} \right.[/tex]
Now, the limits from either side are,
[tex]\lim_{x \to 0^-} f(x) = \lim_{x \to 0} (5x-8) = -8[/tex]
[tex]\lim_{x \to 0^+} f(x) = \lim_{x \to 0} |-4-x| = \lim_{x \to 0} |x+4|=|4| = 4[/tex]
Since one side of the limits doesn't match, so the limit as x→0 does not exist.
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