Thread: What am I doing wrong in this limit calculation?

Hello everyone,

I have tried to calculate the following limit of a function, but my result doesn't match the one that my computer gives me.



First I multiply by , which gives me



Next I simplify the numerator and factor x out of the roots in the denominator



x cancels out, which gives me



My computer algebra system however tells me that the limit is 1/2. It would be -1/2 if x were approaching positive infinity, so I my problem is somehow related to x approaching negative infinity. But what exactly is it that I am doing wrong?

For positive, the function is defined as the unique positive real number such that . Hence, when is negative, . The limit is taken when tends to . Thus, in the denominator you must extract instead of in each one of the square roots.

Thank you for your help! It does work, when I factor -x out of the square root instead of x.

However there is still one more thing that I would like to ask in reference to this problem. For a limit with x approaching a negative number, when do I have to factor -x out of a square root and when do I have to factor out x?
The reason I ask is, because for factoring out -x would be wrong, since this function converges to positive infinity and not to negative infinity.

You are welcome!

Simply apply (this works always). In your last question, when , and tends to as

Right, when , replacing the factored out |x| with -x should work, since as .
Thanks once again.