Question
the vertical asymptotes (if any) of the graph of the function. (Enter your answers as a comma-separated list. If an answer does not exis \( g(x)=\frac{\frac{1}{2} x^{3}-5 x^{2}+12 x}{5 x^{2}-50 x+120} \) \( x \)
Ask by Cervantes Wagner. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The function has no vertical asymptotes.
Solution
Function by following steps:
- step0: Find the vertical asymptotes:
\(g\left(x\right)=\frac{\frac{1}{2}x^{3}-5x^{2}+12x}{5x^{2}-50x+120}\)
- step1: Evaluate the limit:
\(\begin{align}&\lim _{x\rightarrow 4}\left(\frac{\frac{1}{2}x^{3}-5x^{2}+12x}{5x^{2}-50x+120}\right)\\&\lim _{x\rightarrow 6}\left(\frac{\frac{1}{2}x^{3}-5x^{2}+12x}{5x^{2}-50x+120}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&\frac{2}{5}\\&\frac{3}{5}\end{align}\)
- step3: \(x=4\) is not a vertical asymptote\(:\)
\(\begin{align}&x=4\textrm{ }\textrm{is not a vertical asymptote}\\&x=6\textrm{ }\textrm{is not a vertical asymptote}\end{align}\)
- step4: The function has no vertical asymptote:
\(\textrm{No vertical asymptotes}\)
The function \( g(x) = \frac{\frac{1}{2} x^{3}-5 x^{2}+12 x}{5 x^{2}-50 x+120} \) has no vertical asymptotes.
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Simplify this solution Extra Insights
To find the vertical asymptotes of the function \( g(x)=\frac{\frac{1}{2} x^3 - 5 x^2 + 12 x}{5 x^2 - 50 x + 120} \), we need to set the denominator equal to zero and solve for \( x \). The denominator is \( 5 x^2 - 50 x + 120 \). Factoring out a 5, we get \( 5(x^2 - 10x + 24) \). Now we need to solve for \( x \) in the quadratic equation \( x^2 - 10x + 24 = 0 \). Using the quadratic formula, we find the roots are \( x = 6 \) and \( x = 4 \). Thus, the vertical asymptotes occur at \( x = 6 \) and \( x = 4 \). The vertical asymptotes of the graph of the function are located at \( x = 4, 6 \).
