I came across a question in a book asking for the asymptotes of y=1+x2. The answer says it as 2, but I am pretty convinced that there is no asymptote. I somehow think the book is wrong. Any help would be much appreciated.

Edward Hendricks 2022-08-14 Answered
Asymptotes. Calculus.
I came across a question in a book asking for the asymptotes of y = 1 + x 2 . The answer says it as 2, but I am pretty convinced that there is no asymptote. I somehow think the book is wrong. Any help would be much appreciated.
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Answers (2)

Holly Crane
Answered 2022-08-15 Author has 14 answers
You're right that it doesn't have vertical asymptotes anywhere, but it does have 2 oblique asymptotes. As x + ,
1 + x 2 = x 1 x 2 + 1 ,  which approaches  0 + 1 = x , which approaches 1 + x 2 = x 1 x 2 + 1 ,  which approaches  0 + 1 = x ,
since for large x 1 x 2 0. This shows that the line y=x is an oblique asymptote (since the function is getting closer and closer to x). When x , you can almost use the same argument, except you have to pull out a −x, so the line y=−x is also an oblique asymptote.
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Jazmin Clark
Answered 2022-08-16 Author has 5 answers
The book is correct, it has two oblique asymptotes. To show this, write y=ax+b, then:
a = lim x +   f ( x ) x     a n d     b = lim x +   f ( x ) a x
So taking the limit as x + we have
a = lim x +   1 + x 2 x = 1
and
b = lim x +   1 + x 2 x = 0
So you have y=x when x .
Its up to you to find y = x when x
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