 # 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 $y=\sqrt{1+{x}^{2}}$. 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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You're right that it doesn't have vertical asymptotes anywhere, but it does have 2 oblique asymptotes. As $x\to +\mathrm{\infty }$,
which approaches
since for large x $\frac{1}{{x}^{2}}\approx 0$. This shows that the line y=x is an oblique asymptote (since the function is getting closer and closer to x). When $x\to -\mathrm{\infty }$, 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.
###### Not exactly what you’re looking for? Jazmin Clark
The book is correct, it has two oblique asymptotes. To show this, write y=ax+b, then:

So taking the limit as $x\to +\mathrm{\infty }$ we have

and

So you have y=x when $x\to \mathrm{\infty }$.
Its up to you to find $y=-x$ when $x\to -\mathrm{\infty }$