lim_(x rarr0)(4x^(3)-2x^(2)+x)/(3x^(2)+2x)

Identify Problem Type: Identify the type of limit problem. We are dealing with a rational function where both the numerator and the denominator are polynomials. As x approaches 0, both the numerator and the denominator approach 0, which is an indeterminate form 0/0. We need to simplify the expression or use L'Hôpital's Rule to find the limit.Simplify by Factoring: Simplify the expression by factoring if possible. We can factor out an x from both the numerator and the denominator to simplify the expression. (4x^3 - 2x^2 + x) / (3x^2 + 2x) = x(4x^2 - 2x + 1) / x(3x + 2) Now we can cancel out the common factor of x from the numerator and the denominator, provided x is not equal to 0. (4x^2 - 2x + 1) / (3x + 2)Evaluate Limit: Evaluate the limit of the simplified expression as x approaches 0. lim_(x -> 0) (4x^2 - 2x + 1) / (3x + 2) Substitute x = 0 into the simplified expression. (4(0)^2 - 2(0) + 1) / (3(0) + 2) = 1 / 2State Final Answer: State the final answer. The limit of the function (4x^3 - 2x^2 + x) / (3x^2 + 2x) as x approaches 0 is 1/2.

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lim_(x rarr0)(4x^(3)-2x^(2)+x)/(3x^(2)+2x)

$\lim _{x \rightarrow 0} \frac{4 x^{3}-2 x^{2}+x}{3 x^{2}+2 x}$

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Algebra 2

Sum of finite series not start from 1

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\lim _{x \rightarrow 0} \frac{4 x^{3}-2 x^{2}+x}{3 x^{2}+2 x}

Identify Problem Type: Identify the type of limit problem. $\newline$ We are dealing with a rational function where both the numerator and the denominator are polynomials. As $x$ approaches $0$ , both the numerator and the denominator approach $0$ , which is an indeterminate form $0/0$ . We need to simplify the expression or use L'Hôpital's Rule to find the limit.
Simplify by Factoring: Simplify the expression by factoring if possible. $\newline$ We can factor out an $x$ from both the numerator and the denominator to simplify the expression. $\newline$ $\frac{4x^3 - 2x^2 + x}{3x^2 + 2x} = \frac{x(4x^2 - 2x + 1)}{x(3x + 2)}$ $\newline$ Now we can cancel out the common factor of $x$ from the numerator and the denominator, provided $x$ is not equal to $0$ . $\newline$ $\frac{4x^2 - 2x + 1}{3x + 2}$
Evaluate Limit: Evaluate the limit of the simplified expression as $x$ approaches $0$ . $\lim_{x \to 0} \frac{4x^2 - 2x + 1}{3x + 2}$ Substitute $x = 0$ into the simplified expression. $\frac{4(0)^2 - 2(0) + 1}{3(0) + 2} = \frac{1}{2}$
State Final Answer: State the final answer. $\newline$ The limit of the function $(4x^3 - 2x^2 + x) / (3x^2 + 2x)$ as $x$ approaches $0$ is $1/2$ .