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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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.

Holly Crane · Accepted Answer

You&#039;re right that it doesn&#039;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.

Jazmin Clark · Accepted Answer

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  xSo taking the limit as   x  →  +  ∞ we have  a  =      lim          x      →      +      ∞                     1      +              x        2              x    =  1and   b  =      lim          x      →      +      ∞               1    +          x      2        −  x  =  0So you have y=x when   x  →  ∞.Its up to you to find   y  =  −  x when   x  →  −  ∞

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.

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