How do you find all the asymptotes for function f(x)=[(5x+3)/(2x-3)]+1?

Mar 8, 2016

$\textcolor{g r e e n}{\text{Horizontal asymptote at } y = \frac{5}{2}}$

$\textcolor{g r e e n}{\text{Vertical asymptote at } x = \frac{3}{2}}$

Explanation:

The objective of this type of question is to make you think about what an equation is actually doing or saying about a situation.

Asymptotes are where the equation is getting very close to a value that is undefined or a value that does indeed exists but only as $x = \text{infinity or zero}$ is approached.

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There are two things to bear in mind with your question.

$\textcolor{b r o w n}{\text{Point 1.}}$ It contains a fraction. Thus the denominator is undefined
$\textcolor{w h i t e}{\ldots \ldots \ldots . .}$(not allowed) to become zero.

color(brown)("Point 2.")" It will have a specific value that it tends to as "x $\textcolor{w h i t e}{\ldots \ldots \ldots \ldots .} \text{ becomes enormously large}$

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$\textcolor{b l u e}{\text{Consider point 1}}$

$2 x - 3 = 0 \implies x = \frac{3}{2}$

$\textcolor{g r e e n}{\text{Vertical asymptote at } x = \frac{3}{2}}$
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$\textcolor{b l u e}{\text{Consider point 2}}$

As $x$ becomes extremely large the equation will behave as if it is $\left(\frac{5}{2} \times \frac{x}{x}\right) + 1 \text{ " =" " (5/2xx1)+1" " = " } \frac{7}{2}$

$\textcolor{g r e e n}{\text{Horizontal asymptote at } y = \frac{5}{2}}$

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