Dunn Chen
08/11/2024 · Senior High School
3. Se define la función \( f: \operatorname{Dom}(f) \rightarrow I R \) por \[ f(x)=\frac{x^{3}-25 x}{x^{2}-5 x-50} \] Determine TODAS las as' intotas de la función
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Find the vertical asymptotes of \( (x^3-25*x)/(x^2-5*x-50) \).
Function by following steps:
- step0: Find the vertical asymptotes:
\(y=\frac{x^{3}-25x}{x^{2}-5x-50}\)
- step1: Evaluate the limit:
\(\begin{align}&\lim _{x\rightarrow -5}\left(\frac{x^{3}-25x}{x^{2}-5x-50}\right)\\&\lim _{x\rightarrow 10}\left(\frac{x^{3}-25x}{x^{2}-5x-50}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&-\frac{10}{3}\\&\textrm{The limit does not exist}\end{align}\)
- step3: \(x=-5\) is not a vertical asymptote\(:\)
\(\begin{align}&x=-5\textrm{ }\textrm{is not a vertical asymptote}\\&x=10\textrm{ }\textrm{is a vertical asymptote}\end{align}\)
- step4: List all vertical asymptotes of the function:
\(\begin{align}&x=10\end{align}\)
Find the horizontal asymptotes of \( (x^3-25*x)/(x^2-5*x-50) \).
Function by following steps:
- step0: Find the horizontal asymptotes:
\(y=\frac{x^{3}-25x}{x^{2}-5x-50}\)
- step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(y\right)\) and \(\lim _{x\rightarrow -\infty}\left(y\right):\)
\(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{x^{3}-25x}{x^{2}-5x-50}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{x^{3}-25x}{x^{2}-5x-50}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&+\infty\\&-\infty\end{align}\)
- step3: The function has no horizontal asymptotes:
\(\textrm{No horizontal asymptotes}\)
Find the oblique asymptotes of \( (x^3-25*x)/(x^2-5*x-50) \).
Function by following steps:
- step0: Find the oblique asymptotes:
\(y=\frac{x^{3}-25x}{x^{2}-5x-50}\)
- step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(\frac{y}{x}\right)\) and \(\lim _{x\rightarrow -\infty}\left(\frac{y}{x}\right):\)
\(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{\frac{x^{3}-25x}{x^{2}-5x-50}}{x}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{\frac{x^{3}-25x}{x^{2}-5x-50}}{x}\right)\end{align}\)
- step2: Calculate:
\(\begin{align}&1\\&1\end{align}\)
- step3: Evaluate the limit \(\lim _{x\rightarrow +\infty}\left(y-ax\right):\)
\(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{x^{3}-25x}{x^{2}-5x-50}-x\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{x^{3}-25x}{x^{2}-5x-50}-x\right)\end{align}\)
- step4: Calculate:
\(\begin{align}&5\\&5\end{align}\)
- step5: Write the oblique asymptote according to limits value:
\(\begin{align}&y=x+5\textrm{ }\textrm{is an oblique asymptote}\\&y=x+5\textrm{ }\textrm{is an oblique asymptote}\end{align}\)
- step6: The function has one oblique asymptote:
\(y=x+5\)
La función \( f(x)=\frac{x^{3}-25 x}{x^{2}-5 x-50} \) tiene la siguiente información sobre sus asintotas:
- Asintota vertical: \( x=10 \)
- No hay asintotas horizontales.
- Asintota oblicua: \( y=x+5 \)
Quick Answer
La función tiene una asintota vertical en \( x=10 \), no tiene asintotas horizontales y una asintota oblicua en \( y=x+5 \).
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