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

The limit of this function does not exist.

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) \\eq \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.