Question

Use properties of limits and algebraic methods to find the limit, if it exists. (If the limit is infinite, enter '[infinity]' or '-[infinity]', as appropriate. If the limit does not otherwise exist, enter DNE.) lim x→3 f(x),
where f(x) = 9 − 3x if x < 3 ;

and x^2 − x if x ≥ 3

Answer 1

Answer:

The limit of this function does not exist.

Step-by-step explanation:

$\lim_{x \to 3} f(x)$

$f(x)=\left \{ {{9-3x} \quad if \>{x \>< \>3} \atop {x^{2}-x }\quad if \>{x\ \geq \>3 }} \right.$

To find the limit of this function you always need to evaluate the one-sided limits. In mathematical language the limit exists if

$\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) =L$

and the limit does not exist if

$\lim_{x \to a^{-}} f(x) \neq \lim_{x \to a^{+}} f(x)$

Evaluate the one-sided limits.

The left-hand limit

$\lim_{x \to 3^{-} } 9-3x= \lim_{x \to 3^{-} } 9-3*3=0$

The right-hand limit

$\lim_{x \to 3^{+} } x^{2} -x= \lim_{x \to 3^{+} } 3^{2}-3 =6$

Because the limits are not the same the limit does not exist.