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Sagot :
Given:
The function is
[tex]f(x)=\dfrac{x^2-2x-2}{x-2}[/tex]
To find:
The vertical asymptote and oblique asymptote.
Solution:
We have,
[tex]f(x)=\dfrac{x^2-2x-2}{x-2}[/tex]
To find vertical asymptote, equate denominator equal to 0.
[tex]x-2=0[/tex]
[tex]x=2[/tex]
So, the vertical asymptote is [tex]x=2[/tex].
In the given function degree of numerator is greater than denominator so, their is an oblique asymptote. To find oblique asymptote divide the numerator by denominator.
Dividing [tex]x^2-2x-2[/tex] by [tex]x-2[/tex] using synthetic division, we get
2 | 1 -2 -2
2 0
--------------------------
1 0 -2
-------------------------
Here, starting elements of bottom row represent coefficient of quotient and last element of bottom row represents the remainder.
[tex]Quotient=x, Remainder=-2[/tex]
Since, quotient is x, therefore, the oblique asymptote is [tex]y=x[/tex].
Therefore, the correct option is B.
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