- #36
Jameson
Gold Member
MHB
- 4,541
- 13
(Ok, here is my last post on this for now. I think you need to see the next few steps written out so here they are plus some advice)
I think you might want to practice some easier problems first for using these methods. This one has a lot of parts to it if you're seeing it for the first time. Usually limits are given where you have rationalize a fraction somehow and then everything will cancel nicely and easily. After that the cancellation might be trickier. Finally limits like #3 are given and it can be messy.
Being able to quickly choose the proper fraction to multiply with the original limit is key. It is usually going to be the conjugate of the numerator or denominator. Here are some examples.
This is where we are at for #3 so far:
From here you need to divide all terms by $n$. That means evaluate
For
You hopefully see now why algebra is so important here. Review these concepts some and practice a few easier problems to get a feel for the process. Once you have that you can do this problem in 1-2 minutes probably by knowing the methods you'll probably use.
I think you might want to practice some easier problems first for using these methods. This one has a lot of parts to it if you're seeing it for the first time. Usually limits are given where you have rationalize a fraction somehow and then everything will cancel nicely and easily. After that the cancellation might be trickier. Finally limits like #3 are given and it can be messy.
Being able to quickly choose the proper fraction to multiply with the original limit is key. It is usually going to be the conjugate of the numerator or denominator. Here are some examples.
This is where we are at for #3 so far:
\(\displaystyle \frac{n\left(\sqrt{n^2+4}-n \right) \left(\sqrt{n^2+4}+n\right)}{\sqrt{n^2+4}+n}=\frac{4n}{\sqrt{n^2+4}+n}\)
From here you need to divide all terms by $n$. That means evaluate
\(\displaystyle \frac{4n}{n}, \frac{\sqrt{n^2+4}}{n},\frac{n}{n}\)
\(\displaystyle \frac{\sqrt{n^2+4}}{n}\) you can use this property: \(\displaystyle \frac{\sqrt{n^2+4}}{n}=\sqrt{\frac{n^2+4}{n^2}} =\sqrt{ \frac{n^2}{n^2}+\frac{4}{n^2}}\)
You hopefully see now why algebra is so important here. Review these concepts some and practice a few easier problems to get a feel for the process. Once you have that you can do this problem in 1-2 minutes probably by knowing the methods you'll probably use.