Limits question, solving radical multiplication

[Bmanmcfly]

Hi, I've looked through my books and having been out of school a few years, I need a bit of help with a limit problem...

Im familiar with conjugating to handle radicals, but I can't seem to figure out how to handle an example where the radical is only multiplied...

(X²+5)sqrt(x²-4)

I just need the first few steps to figure out what I would multiply top and bottom when it's not really a difference of squares situation... At least not like the other examples I've been finding that seems to give a wrong answer?

Thanks for any help.

 

[Yogi]

Where is the limit part of this limit problem?

 

[Bmanmcfly]

The lim is x->1, but it's somewhat irrelevant, I'm just not sure how I would rationalize / conjugate in the situation of the limit of a multiplied radical.

If the question was
(X²+5)+/-sqrt(x²-4) I get it, but when it's a multiplication I'm doing something wrong... The lim value is one where the function would be indeterminate initially.

 

[soroban]

Hello, Bmanmcfly!

Why are you worried about conjugates?
Did you try to evaluate the limit?

\(\displaystyle \displaystyle \lim_{x\to1}\,(x^2+5)\sqrt{x^2-4}\)

\(\displaystyle \displaystyle \lim_{x\to1}\,(x^2+5)\sqrt{x^2-4} \;=\; (1^2+5)\sqrt{1^2-4} \;=\;6\sqrt{-3} \;=\;6i\sqrt{3}\)

If complex values are allowed, that is the limit.

 

[HallsofIvy]

And, of course, if complex values are not allowed, you wouldn't be able to take the limit "as x goes to 1". In terms of real numbers, \(\displaystyle \sqrt{x^2- 4}\) is not defined between -2 and 2.