To find the limit of the function [tex]\( f(x) = 3x^3 - 2x + 7 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex], we will follow these steps:
1. Substitute [tex]\( x = -2 \)[/tex] into the function:
- First, evaluate [tex]\( x^3 \)[/tex] where [tex]\( x = -2 \)[/tex]:
[tex]\[
(-2)^3 = -8
\][/tex]
- Next, multiply the result by 3:
[tex]\[
3 \cdot (-8) = -24
\][/tex]
2. Substitute [tex]\( x = -2 \)[/tex] into the linear term involving [tex]\( x \)[/tex]:
- Multiply [tex]\(-2\)[/tex] by [tex]\(-2\)[/tex]:
[tex]\[
-2 \cdot (-2) = 4
\][/tex]
3. Add the constant term 7:
- Combine the results of the previous steps:
[tex]\[
-24 + 4 + 7 = -13
\][/tex]
Therefore, the limit of the function as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] is:
[tex]\[
\lim _{x \rightarrow -2}\left(3 x^3 - 2 x + 7\right) = -13
\][/tex]
