I have the function: f ( x ) = 2 x &#x2212;<!-- − --> 2 <mrow

Finley Mckinney

Finley Mckinney

Answered question

2022-06-27

I have the function: f ( x ) = 2 x 2 x + 2
I know that this function has an oblique asymptote, but all the tutorials I can find on google, are with rational functions with the form:
f ( x ) = P ( x ) Q ( x )
Where they simply just divide the denominator with the numerator.
But I can't do that, because my equation doesn't contain any fractions. So my question is: how do I find the function to the oblique asymptote for my f ( x )?

Answer & Explanation

Samantha Reid

Samantha Reid

Beginner2022-06-28Added 22 answers

Hint: for   x ,   2 x 0
the oblique asymptote has the form    y = m x + q
and to find m and q you calculate the following limits:
  m = lim x   f ( x ) / x
  q = lim x   [ f ( x ) m x ]
excluderho

excluderho

Beginner2022-06-29Added 8 answers

An asymptote is a line that approximates the function as x ± . In other words, you want a function of the form f ( x ) = a x + b such that
lim x ± 2 x 2 x + 2 a x + b = 1
This is not possible for x + since the 2 x explodes, so you could have an asymptote only for x :
lim x 2 x 2 x + 2 a x + b = lim x ( 2 x + 2 a x + b 2 x a x + b ) = 1
Since lim x 2 x a x + b = 0, you are left with
lim x 2 x + 2 a x + b = 1
which yelds a = 2, b = 2

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