Question

Please help me simplify the following...

Answer 1

\[\frac{ \sqrt{(x+h+1)}-\sqrt{x+1} }{ h }\]

Answer 2

ok... you need to rationalise the numerator

so multiply by \[\frac{ \sqrt{x + h + 1} + \sqrt{x + 1}}{\sqrt{x + h + 1}+\sqrt{x + 1}}\]

Answer 3

You can't really simplify it much, but it looks like a derivative, are you sure there's no missing information?

Answer 4

lim
h--0

Answer 5

should be in front, yes it is a derivative

Answer 6

\[Z=\frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}\]\[Z*h=\sqrt{x+h+1}-\sqrt{x+1}\]\[Z*h*(\sqrt{x+h+1}+\sqrt{x+1})=(\sqrt{x+h+1}-\sqrt{x+1})*(\sqrt{x+h+1}+\sqrt{x+1})\]\[=(x+h+1)-(x+1)=h\]\[Z*(\sqrt{x+h+1}+\sqrt{x+1})=1\]\[Z=\frac{1}{\sqrt{x+h+1}+\sqrt{x+1}}\]

Answer 7

Okay, then it's just the derivative of
\[ f(x) = \sqrt{x+1} \]
which is
\[ f'(x) = \lim_{h->0}{\frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}} = \frac{1}{2\sqrt{1+x}} \]

Answer 8

so you'll end up with

\[\lim_{h \rightarrow 0} \frac{(\sqrt{x + h + 1}-\sqrt{x +1})(\sqrt{x + h +1}+\sqrt{x +1}}{h(\sqrt{x + h + 1}+\sqrt{x + 1})}\]

all you need to do is simplify

the numerator is now the difference of 2 squares...

Answer 9

the input the value for h to find the limit

Answer 10

\[\frac{ h }{ h(\sqrt{x+h+1}+\sqrt{x+1)} }\]

Answer 11

am i on the right track

Answer 12

thats it,...well done

Answer 13

ok, do i cancel the h's?

Answer 14

\[\lim_{h\rightarrow 0}\frac{1}{(\sqrt{x+h+1}+\sqrt{x+1})}=\frac{1}{(\sqrt{x+1}+\sqrt{x+1})}=\frac{1}{2\sqrt{x+1}}\]

Answer 15

yes...

the for the h in the denominator apply the limit...

i.e. substitute h = 0

Answer 16

that will give

\[\frac{1}{\sqrt{x + 1} + \sqrt{x + 1}}\]

Answer 17

\[\frac{ 1 }{ 2\sqrt{x+1} }\]

Answer 18

@dape you can use \rightarrow for ->\[\rightarrow\]

Answer 19

thats correct, well done.. hope we helped

Answer 20

thank you, yes very much