S5.t.5 Find domain and asymptotes.

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Using polynomial long division, we can rewrite the rational function as $2 + \dfrac{-20x+64}{x^2+6x-40}$. As $x$ approaches infinity, the fraction $\dfrac{-20x+64}{x^2+6x-40}$ approaches $0$, making the limit equal to $2$. Therefore, the horizontal asymptote is $y=2$.
  • #1
karush
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$\tiny{s5.t.5 Kiaser HS}$
Find domain asymptotes.
$g(x)=\dfrac{2x^2-14x+24}{x^2+6x-40}$
$\begin{array}{rll}
\textsf{factor}&=\dfrac{2(x-3)(x-4)}{(x-4)(x+10)}
=\dfrac{2(x-3)\cancel{(x-4)}}{\cancel{(x-4)}(x+10)}
=\dfrac{2(x-3)}{x+10}\\
\textsf{Domain} & -\infty<-10<\infty\\
HA \quad y&=3 \\
VA \quad x&=-10
\end{array}$
I think there is an oblique asymptote but ??
Also the OP has a hole at $x=4$ but they didn't ask for it ?

btw how come tab does not work here?:confused:
 
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  • #2
what does $-\infty < -10 < \infty$ mean?
the domain is all reals except x = 4 and x = -10 …
$x \in (-\infty, -10) \cup (-10,4) \cup (4, \infty)$

horizontal asymptote is y = 2

oblique asymptotes occur when the degree of the numerator is one greater than the degree of the denominator

yes, the original rational function has a removable discontinuity at x = 4
 
  • #3
how do you get y=2 for HA
 
  • #4
karush said:
how do you get y=2 for HA

$\displaystyle \lim_{x \to \pm \infty} \dfrac{2x^2-14x+24}{x^2+6x-40}$
 

FAQ: S5.t.5 Find domain and asymptotes.

What is the domain of a function?

The domain of a function is the set of all possible input values for the function. It is the set of values that the independent variable can take on in order for the function to be defined.

How do you find the domain of a function?

To find the domain of a function, you need to look for any restrictions on the independent variable. This can include things like division by zero, square roots of negative numbers, or logarithms of non-positive numbers. The domain will be all real numbers that do not violate these restrictions.

What is an asymptote?

An asymptote is a line that a graph approaches but never touches. It can be horizontal, vertical, or oblique (slanted).

How do you find the asymptotes of a function?

To find the asymptotes of a function, you need to look for any values of the independent variable that make the function undefined. These values will correspond to vertical asymptotes. You also need to check for any horizontal or oblique asymptotes by taking the limit of the function as the independent variable approaches positive or negative infinity.

Why is it important to find the domain and asymptotes of a function?

Knowing the domain and asymptotes of a function is important because it helps us understand the behavior of the function and how it relates to the real world. It also allows us to identify any potential errors or limitations in our calculations or interpretations of the function.

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