To solve the limit [tex]\(\lim _{x \rightarrow -2} \frac{3 x^2-2 x+5}{2 x+1}\)[/tex], let's follow a step-by-step approach.
1. Substitution Approach:
First, we'll try to substitute [tex]\( x = -2 \)[/tex] directly into the expression to see if we get a determinate form:
[tex]\[
\text{Numerator: } 3(-2)^2 - 2(-2) + 5 = 3 \cdot 4 + 4 + 5 = 12 + 4 + 5 = 21
\][/tex]
[tex]\[
\text{Denominator: } 2(-2) + 1 = -4 + 1 = -3
\][/tex]
So, substituting [tex]\( x = -2 \)[/tex] directly, we have:
[tex]\[
\frac{21}{-3} = -7
\][/tex]
2. Conclusion:
We see that substituting [tex]\( x = -2 \)[/tex] directly into the expression [tex]\(\frac{3 x^2 - 2 x + 5}{2 x + 1}\)[/tex] gives us a determinate value of:
[tex]\[
\frac{21}{-3} = -7
\][/tex]
Thus, we conclude that the limit is:
[tex]\[
\lim _{x \rightarrow -2} \frac{3 x^2 - 2 x + 5}{2 x + 1} = -7
\][/tex]
