Question
b) \( \lim _{x \rightarrow 0} \frac{6 x}{x^{3}-6 x^{2}+9 x} \)
Ask by Elliott Mullins. in Greece
Nov 21,2024
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Tutor-Verified Answer
Answer
El límite de \( \frac{6x}{x^{3}-6x^{2}+9x} \) cuando \( x \) se aproxima a 0 es \( \frac{2}{3} \) o aproximadamente 0.6.
Solution
Calculate the limit \( \lim _{x \rightarrow 0} \frac{6 x}{x^{3}-6 x^{2}+9 x} \).
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
\(\lim _{x\rightarrow 0}\left(\frac{6x}{x^{3}-6x^{2}+9x}\right)\)
- step1: Reduce the fraction:
\(\lim _{x\rightarrow 0}\left(\frac{6}{\left(x-3\right)^{2}}\right)\)
- step2: Rewrite the expression:
\(\frac{\lim _{x\rightarrow 0}\left(6\right)}{\lim _{x\rightarrow 0}\left(\left(x-3\right)^{2}\right)}\)
- step3: Evaluate:
\(\frac{6}{\lim _{x\rightarrow 0}\left(\left(x-3\right)^{2}\right)}\)
- step4: Evaluate:
\(\frac{6}{9}\)
- step5: Reduce the fraction:
\(\frac{2}{3}\)
The limit of \( \frac{6x}{x^{3}-6x^{2}+9x} \) as \( x \) approaches 0 is equal to \( \frac{2}{3} \) or approximately 0.6.
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Simplify this solution Beyond the Answer
To evaluate the limit \( \lim_{x \rightarrow 0} \frac{6 x}{x^{3} - 6 x^{2} + 9 x} \), we first simplify the expression. We can factor the denominator: \[ x^{3} - 6 x^{2} + 9 x = x(x^2 - 6x + 9) = x(x - 3)^2 \] Now, substituting the factored form into the limit gives us: \[ \lim_{x \rightarrow 0} \frac{6x}{x(x - 3)^2} \] We can cancel \( x \) (noting that we don't need to worry about \( x = 0 \) since it’s just the limit): \[ \lim_{x \rightarrow 0} \frac{6}{(x - 3)^2} \] Now, we can substitute \( x = 0 \): \[ \frac{6}{(0 - 3)^2} = \frac{6}{9} = \frac{2}{3} \] Thus, the limit is: \[ \boxed{\frac{2}{3}} \]
