lim_(x rarr3)(x^(3)-9x)/(x^(2)-3x)=

Identify Limit: Identify the limit to be calculated. We need to find the limit of the function (x^3 - 9x) / (x^2 - 3x) as x approaches 3.Direct Substitution: Direct substitution to check for indeterminate forms. Substitute x = 3 into the function to see if the limit can be calculated directly. lim_(x -> 3)(x^3 - 9x) / (x^2 - 3x) = (3^3 - 9*3) / (3^2 - 3*3) = (27 - 27) / (9 - 9) = 0 / 0 We have an indeterminate form, so we cannot calculate the limit by direct substitution.Factor Expression: Factor the numerator and denominator to simplify the expression. Factor out the common terms in the numerator and the denominator. Numerator: x^3 - 9x = x(x^2 - 9) = x(x + 3)(x - 3) Denominator: x^2 - 3x = x(x - 3)Cancel Common Factors: Cancel out the common factors. The term (x - 3) is common in both the numerator and the denominator and can be canceled out. lim_(x -> 3)(x(x + 3)(x - 3)) / (x(x - 3)) = lim_(x -> 3)(x + 3)Calculate Simplified Limit: Calculate the limit of the simplified expression. Now that the expression is simplified, we can substitute x = 3 directly. lim_(x -> 3)(x + 3) = 3 + 3 = 6

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$\lim _{x \rightarrow 3} \frac{x^{3}-9 x}{x^{2}-3 x}=$

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Algebra 2

Composition of linear and quadratic functions: find a value

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\lim _{x \rightarrow 3} \frac{x^{3}-9 x}{x^{2}-3 x}=

Identify Limit: Identify the limit to be calculated. $\newline$ We need to find the limit of the function $(x^3 - 9x) / (x^2 - 3x)$ as $x$ approaches $3$ .
Direct Substitution: Direct substitution to check for indeterminate forms. $\newline$ Substitute $x = 3$ into the function to see if the limit can be calculated directly. $\newline$ $\lim_{x \to 3}\frac{x^3 - 9x}{x^2 - 3x} = \frac{3^3 - 9\cdot 3}{3^2 - 3\cdot 3}$ $\newline$ $= \frac{27 - 27}{9 - 9}$ $\newline$ $= \frac{0}{0}$ $\newline$ We have an indeterminate form, so we cannot calculate the limit by direct substitution.
Factor Expression: Factor the numerator and denominator to simplify the expression. $\newline$ Factor out the common terms in the numerator and the denominator. $\newline$ Numerator: $x^3 - 9x = x(x^2 - 9) = x(x + 3)(x - 3)$ $\newline$ Denominator: $x^2 - 3x = x(x - 3)$
Cancel Common Factors: Cancel out the common factors. $\newline$ The term $(x - 3)$ is common in both the numerator and the denominator and can be canceled out. $\newline$ $\lim_{x \to 3}\frac{x(x + 3)(x - 3)}{x(x - 3)} = \lim_{x \to 3}(x + 3)$
Calculate Simplified Limit: Calculate the limit of the simplified expression. $\newline$ Now that the expression is simplified, we can substitute $x = 3$ directly. $\newline$ $\lim_{x \to 3}(x + 3) = 3 + 3 = 6$