# Thread: fiNd the VaLue of

1. ## fiNd the VaLue of

Find the value of $\displaystyle \lim(n\rightarrow 4)(\frac{n-4}{\sqrt{n}-2})$.

^include working and explanation needed pls

2. Dear mathbuoy,

use this:
$\displaystyle n - 4 = (\sqrt 2)^2 - 2^2$
and
$\displaystyle a^2 - b^2 = (a+b)*(a-b)$

3. Hi

Originally Posted by mathbuoy
Find the value of $\displaystyle \lim(n\rightarrow 4)(\frac{n-4}{\sqrt{n}-2})$.

^include working and explanation needed pls
$\displaystyle lim_{n \to 4} \sqrt{n}-2 = 0$
$\displaystyle lim_{n \to 4} n - 4 = 0$

L'Hospital:

$\displaystyle lim_{n \to 4} \frac{n-4}{\sqrt{n}-2} =$
$\displaystyle lim_{n \to 4} \frac{1}{(\sqrt{n})' } = lim_{n \to 4} \frac{1}{0.5 * n^{-0.5}}$

$\displaystyle = lim_{n \to 4} n^{0.5}/0.5 =lim_{n \to 4} n^{0.5}*2 = 4^{0.5} * 2 = 4$

4. Originally Posted by mathbuoy
Find the value of $\displaystyle \lim(n\rightarrow 4)(\frac{n-4}{\sqrt{n}-2})$.

^include working and explanation needed pls

$\displaystyle \lim_{n \to 4} \frac{n-4}{\sqrt{n} -2 }$

Multiply by conjugate

$\displaystyle \lim_{n \to 4} \bigg(\frac{n-4}{\sqrt{n} -2 }\bigg)\bigg(\frac{\sqrt{n} + 2 }{\sqrt{n} + 2}\bigg)$

$\displaystyle \lim_{n \to 4} \frac{(n-4)(\sqrt{n} + 2)}{n-4}$

$\displaystyle \lim_{n \to 4} \sqrt{n} +2$

4

5. With Skalkaz hint (btw it has a typo in it)

$\displaystyle lim \frac{n-4}{\sqrt{n} - 2} = lim \frac{(\sqrt{n})^2 - 2^2}{\sqrt{n} - 2}$

$\displaystyle = lim \frac{(\sqrt{n}-2)(\sqrt{n}+2)}{\sqrt{n}-2}$

$\displaystyle = lim_{n -> 4}\sqrt{n}+2 = \sqrt{4} + 2 = 4$

Regards Rapha