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Evaluate the following limit:

[tex]\[ \lim_{x \rightarrow 0} \frac{4x^3 - x^2 + 2x}{3x^2 + 4x} \][/tex]


Sagot :

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]