Answer:
The limit of this function does not exist.
Step-by-step explanation:
[tex]\lim_{x \to 3} f(x)[/tex]
[tex]f(x)=\left \{ {{9-3x} \quad if \>{x \>< \>3} \atop {x^{2}-x }\quad if \>{x\ \geq \>3 }} \right.[/tex]
To find the limit of this function you always need to evaluate the one-sided limits. In mathematical language the limit exists if
[tex]\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) =L[/tex]
and the limit does not exist if
[tex]\lim_{x \to a^{-}} f(x) \neq \lim_{x \to a^{+}} f(x)[/tex]
Evaluate the one-sided limits.
The left-hand limit
[tex]\lim_{x \to 3^{-} } 9-3x= \lim_{x \to 3^{-} } 9-3*3=0[/tex]
The right-hand limit
[tex]\lim_{x \to 3^{+} } x^{2} -x= \lim_{x \to 3^{+} } 3^{2}-3 =6[/tex]
Because the limits are not the same the limit does not exist.
