To find the limit [tex]\(\lim _{x \rightarrow 0} \frac{4x^3 - x^2 + 2x}{3x^2 + 4x}\)[/tex], we will start by examining the given function and carefully analyze its behavior as [tex]\(x\)[/tex] approaches 0.
Given:
[tex]\[
\frac{4x^3 - x^2 + 2x}{3x^2 + 4x}
\][/tex]
First, observe that both the numerator and the denominator have terms that involve [tex]\(x\)[/tex]. It is helpful to factor out the common factors from both the numerator and denominator to simplify the expression.
### Step 1: Factorizing the denominator and numerator:
- Factor [tex]\(x\)[/tex] out of the numerator:
[tex]\[
4x^3 - x^2 + 2x = x(4x^2 - x + 2)
\][/tex]
- Factor [tex]\(x\)[/tex] out of the denominator:
[tex]\[
3x^2 + 4x = x(3x + 4)
\][/tex]
### Step 2: Simplifying the expression:
Substitute these factorizations back into the original limit expression:
[tex]\[
\frac{4x^3 - x^2 + 2x}{3x^2 + 4x} = \frac{x(4x^2 - x + 2)}{x(3x + 4)}
\][/tex]
We can now cancel out the common factor of [tex]\(x\)[/tex] in both the numerator and the denominator:
[tex]\[
\frac{x(4x^2 - x + 2)}{x(3x + 4)} = \frac{4x^2 - x + 2}{3x + 4} \quad \text{for} \quad x \neq 0
\][/tex]
### Step 3: Taking the limit of the simplified expression:
Now, we need to find the limit as [tex]\(x\)[/tex] approaches 0 of the simplified expression:
[tex]\[
\lim _{x \rightarrow 0} \frac{4x^2 - x + 2}{3x + 4}
\][/tex]
Substitute [tex]\(x = 0\)[/tex] into the simplified expression:
- The numerator becomes:
[tex]\[
4(0)^2 - 0 + 2 = 2
\][/tex]
- The denominator becomes:
[tex]\[
3(0) + 4 = 4
\][/tex]
Thus, the limit is:
[tex]\[
\lim _{x \rightarrow 0} \frac{4x^2 - x + 2}{3x + 4} = \frac{2}{4} = \frac{1}{2}
\][/tex]
### Conclusion:
Therefore,
[tex]\[
\lim _{x \rightarrow 0} \frac{4x^3 - x^2 + 2x}{3x^2 + 4x} = \frac{1}{2}
\][/tex]
