Different ways of computing a limit (Part 2)

One of my colleagues placed the following problem on an exam for his Calculus II course…

\displaystyle \lim_{x \to \infty} \frac{\sqrt{x^2+1}}{x}

and was impressed by the variety of correct responses that he received. I thought it would be fun to discuss some of the different ways that this limit can be computed.

Method #2. Using L’Hopital’s Rule. The limit has the indeterminant form \infty/\infty, and so I can differentiate the top and the bottom with respect to x:

\displaystyle \lim_{x \to \infty} \frac{\sqrt{x^2+1}}{x} = \displaystyle \lim_{x \to \infty} \frac{ \displaystyle \frac{d}{dx} \left( \sqrt{x^2+1} \right) }{\displaystyle \frac{d}{dx} \left( x \right)}

= \displaystyle \lim_{x \to \infty} \frac{ \displaystyle \frac{1}{2} \left( x^2+1 \right)^{-1/2} \cdot 2x }{1}

= \displaystyle \lim_{x \to \infty} \frac{x}{\sqrt{x^2+1}}.

Oops… it looks like I just got the reciprocal of the original limit! Indeed, if I use L’Hopital’s Rule again, I’ll just return back to the original limit.

So that doesn’t look very helpful… except it is. If I define the value of this limit to be equal to L, then I’ve just shown that L = 1/L (assuming that the limit exists in the first place, of course). That means that L = 1 or L = -1. Well, clearly the limit of this nonnegative function can’t be negative, and so we conclude that the limit is equal to 1.

Leave a comment

2 Comments

  1. Ooh, yes, that’s a very nice argument.

    Reply
  2. In a similar vein: If you are supposing the limit converges to L in the first place, then its square will converge to L^2.

    But squaring returns \lim_{x to \infty} of 1 + 1/x^2 = 1.

    So L^2 = 1 etc.

    Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: