lim_(x rarr-3)(x^(3)+x^(2)-6x)/(x^(2)+3x)=

Identify Function and Point: Identify the function and the point at which we need to find the limit. We are given the function (x^3 + x^2 - 6x) / (x^2 + 3x) and we need to find its limit as x approaches -3.Simplify if Possible: Simplify the function if possible. We can factor both the numerator and the denominator to simplify the expression. Numerator: x^3 + x^2 - 6x = x(x^2 + x - 6) Denominator: x^2 + 3x = x(x + 3) Now we factor the quadratic x^2 + x - 6 in the numerator. x^2 + x - 6 can be factored into (x + 3)(x - 2). So the function simplifies to: (x(x + 3)(x - 2)) / (x(x + 3))Cancel Common Factors: Cancel out common factors. We can cancel the common factor of x and (x + 3) from the numerator and the denominator. This gives us: (x - 2) / 1 Which simplifies to: x - 2Find Limit: Find the limit by direct substitution. Now that we have simplified the function, we can find the limit by substituting x with -3. (-3) - 2 = -5

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lim_(x rarr-3)(x^(3)+x^(2)-6x)/(x^(2)+3x)=

$\lim _{x \rightarrow-3} \frac{x^{3}+x^{2}-6 x}{x^{2}+3 x}=$

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\lim _{x \rightarrow-3} \frac{x^{3}+x^{2}-6 x}{x^{2}+3 x}=

Identify Function and Point: Identify the function and the point at which we need to find the limit. $\newline$ We are given the function $(x^3 + x^2 - 6x) / (x^2 + 3x)$ and we need to find its limit as $x$ approaches $-3$ .
Simplify if Possible: Simplify the function if possible. $\newline$ We can factor both the numerator and the denominator to simplify the expression. $\newline$ Numerator: $x^3 + x^2 - 6x = x(x^2 + x - 6)$ $\newline$ Denominator: $x^2 + 3x = x(x + 3)$ $\newline$ Now we factor the quadratic $x^2 + x - 6$ in the numerator. $\newline$ $x^2 + x - 6$ can be factored into $(x + 3)(x - 2)$ . $\newline$ So the function simplifies to: $\newline$ $\frac{x(x + 3)(x - 2)}{x(x + 3)}$
Cancel Common Factors: Cancel out common factors. $\newline$ We can cancel the common factor of $x$ and $(x + 3)$ from the numerator and the denominator. $\newline$ This gives us: $\newline$ $\frac{x - 2}{1}$ $\newline$ Which simplifies to: $\newline$ $x - 2$
Find Limit: Find the limit by direct substitution. $\newline$ Now that we have simplified the function, we can find the limit by substituting $x$ with $-3$ . $\newline$ $(-3) - 2 = -5$