find the limit.\nlim (3x³ - 3x² + 2x + 3)\nx→1\nselect the correct choice below and, if necessary, fill in…

find the limit.\nlim (3x³ - 3x² + 2x + 3)\nx→1\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. lim (3x³ - 3x² + 2x + 3) = (simplify your answer.)\nx→1\nb. the limit does not exist.

find the limit.\nlim (3x³ - 3x² + 2x + 3)\nx→1\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. lim (3x³ - 3x² + 2x + 3) = (simplify your answer.)\nx→1\nb. the limit does not exist.

Answer

Explanation:

Step1: Substitute (x = 1) into the polynomial

We use the direct - substitution property for polynomials. For a polynomial (P(x)=3x^{3}-3x^{2}+2x + 3), the limit as (x\rightarrow a) is (P(a)). Substitute (x = 1) into (P(x)=3x^{3}-3x^{2}+2x + 3): [ \begin{align*} P(1)&=3\times(1)^{3}-3\times(1)^{2}+2\times(1)+3\ &=3\times1-3\times1 + 2\times1+3 \end{align*} ]

Step2: Simplify the expression

[ \begin{align*} 3\times1-3\times1 + 2\times1+3&=3 - 3+2 + 3\ &=(3-3)+(2 + 3)\ &=0 + 5 \end{align*} ]

Answer:

A. (\lim_{x\rightarrow1}(3x^{3}-3x^{2}+2x + 3)=5)