Question

Evaluate this limit.

Answer 1

\[\lim_{x \rightarrow 16} \frac{ 16-x }{ 4-\sqrt{x} }\]

Answer 2

I understand that if you just plug in 16 it doesn't work

Answer 3

\[4-\sqrt{16} = 4-4 = 0 \] doesn't work

Answer 4

I'm wondering if I can just say, rationalize the denominator

Answer 5

simplify first??

Answer 6

yes right

Answer 7

\[\frac{ 16-x}{ 4-\sqrt{x} }*\frac{ 4+\sqrt{x} }{ 4+\sqrt{x} }\]

Answer 8

this is what I was thinking

Answer 9

that's correct.

Answer 10

ok thanks =)

Answer 11

that's it ?? can you solve the rest of it ?

Answer 12

\[\frac{ 64+16\sqrt{x}-4x-x \sqrt{x} }{ 16-x }\]

Answer 13

\[\frac{ 64+16*2-4(2)-4*2 }{ 16-4 } = \frac{ 80 }{ 12 }= \frac{ 20 }{ 3 }\]

Answer 14

you don't have to simplify the denominator.
the reason we are doing this bec we don't want 0 at the denominator
so you can just multiply by the conjugate and then just sub in 16 for x :)

Answer 15

wait so how would you do it?

Answer 16

\[\frac{ 16-x}{ 4-\sqrt{x} }*\frac{ 4+\sqrt{x} }{ 4+\sqrt{x} }\]
this is correct
\[\large\rm \frac{ 16-x( 4+\sqrt{x} )}{ (4-\sqrt{x})(4+\sqrt{x})} \rightarrow \frac{ 16-\color{red}{16}( 4+\sqrt{\color{Red}{16}} )}{ (4-\sqrt{\color{red}{16}})(4+\sqrt{\color{ReD}{16}})}\]

Answer 17

oh! wow that's much easier than what I did

Answer 18

I see,you didn't bother to expand anything, just plug in.

Answer 19

yes right. expanding stuff is just kinda mess.

Answer 20

thanks :)

Answer 21

hmmm. i'm sorry i just solved this question and figured out that we do need to simplify the bottom part....

Answer 22

\(\color{blue}{\text{Originally Posted by}}\) @greatlife44
\[\frac{ 64+16\sqrt{x}-4x-x \sqrt{x} }{ 16-x }\]
\(\color{blue}{\text{End of Quote}}\)
hmm this is correct but still if we substitute 16 for x we get 0
you can do this \[\frac{ (16-x)(4+\sqrt{x}) }
{\color{Red}{(4-\sqrt{x})(4+\sqrt{x})} }\]\[\frac{ (16-x)(4+\sqrt{x}) }{ \color{Red}{(16-x) }} \]
\[\frac{ \cancel{(16-x)}(4+\sqrt{x}) }{ \color{Red}{\cancel{(16-x) }}} \]
now sub 16 for x

Answer 23

hi @Nnesha How were you able to write in different colored texts and cross stuff out?

Answer 24

latex :)

Answer 25

`\[\frac{ \cancel{(16-x)}(4+\sqrt{x}) }{ \color{Red}{\cancel{(16-x) }}} \]`
to cancel `\cancel` for color `\color{name}{text}`

Answer 26

:D thanks